Introduction and Summary
Emanuel Swedenborg (1688–1772) addresses all of these questions thoroughly in his theological writings, and provides extensive quotations from the Bible in support of his teachings on these subjects. See especially his small work The Lord and the first three chapters of his major exposition of his theology, True Christianity. (The links are to the listings for these books on the publisher's website.)
In this answer I can provide only a brief version of his teachings on these subjects, and only a few of the supporting scripture references—especially since there are multiple questions, each of which deserves an answer of its own.
In brief, according to Swedenborg, Jesus was indeed God from eternity born into the world, by means of a human nature that he took on from his human mother Mary. During his lifetime on earth he went through a process of "glorification" in which he replaced all of the finite humanity he received from Mary with an infinite divine humanity made of the divine substance of God.
What died on the Cross was the last of his finite humanity from Mary. What rose from death was his fully glorified divine humanity, which was the human manifestation of God. At the time of his Ascension, he became fully one with God, so that Jesus Christ is now simply God, in whom is the Father, the Son, and the Holy Spirit. These are three "essentials" of one God, who is one both in Essence and in Person.
Jesus' primary acts of salvation and redemption were defeating the power of the Devil, or hell, and thereby saving humanity as a whole from the overwhelming power of evil, which was dragging us down to hell. Having gained full power over the Devil, the Lord now keeps humans on earth in a balance between good and evil, and gives us—or really, lends us—the power to defeat the grip of evil on our lives. We humans do not earn our salvation because it is only through the Lord's power working in us that we can be saved. We can therefore take no credit for our own salvation.
According to Swedenborg, there was no "Son born from eternity," as in Nicene Christian belief. Rather, the Son of God came into being at the time that the Lord, who is God from eternity, was born into this world. The Son of God is the same as the divine humanity. The Son of God was initially born of a woman and had a finite human nature, but during Jesus' lifetime on earth it became fully divine, as mentioned above.
This provides a very brief answer to all of the queries posed in this question. Now for a fuller answer to the primary query as presented at the end of the question, including supporting quotations from Swedenborg's writings and from the Bible, and a much briefer answer to the secondary query. First:
Who, according to Swedenborg, is this Jesus, born of a woman, who died and then was resurrected?
In True Christianity #92 Swedenborg writes:
The "Son of God" is the human manifestation in which God sent himself into the world. The Lord frequently says that the Father sent him, or that he was sent by the Father (for example, Matthew 10:40; 15:24; John 3:17, 34; 5:23-24, 36-38; 6:29, 39-40, 44, 57; 7:16, 18, 28-29; 8:16, 18, 29, 42; 9:4; and very often elsewhere). The Lord says this because being sent into the world means coming down among people, which he did through the human manifestation he took on through the Virgin Mary.
The human manifestation really is the Son of God, in that he was conceived by Jehovah God as the Father, as it says in Luke 1:32, 35.
A little later in the same section Swedenborg writes:
In the Lord's case, the divine nature he had came from Jehovah his Father; the human nature he had came from his mother. These two natures united together are "the Son of God." The truth of this is clearly substantiated by the Lord's birth, as recorded in Luke: "The angel Gabriel said to Mary, the Holy Spirit will descend upon you and the power of the Highest will cover you; therefore the Holy One that is born from you will be called the Son of God" (Luke 1:35).
According to Swedenborg, then, when Jesus says that the Father "sent" him into the world, it does not mean that one Person of the Trinity, God the Father, sent another Person of the Trinity, God the Son, into the world, as is believed in Nicene Christianity. Rather it means that God sent himself into the world in a human manifestation, whom the Gospels call Jesus Christ.
It is called "sending" because God did this by fathering a child in the womb of a human woman, the Virgin Mary. That child was "God with us" (Matthew 1:23) because that child's inner self was the eternal God. Because God cannot be divided, Jesus was God as to his inner self even during his lifetime on earth.
Since this is how God came into the world, Jesus had an inner divine being, analogous to our soul, which was God; he also had an outer human self that he received from his human mother Mary, similarly to how humans on earth receive a human nature from their mothers. However, unlike created human beings, Jesus did not receive a human nature from any human father; instead, he received an inner divine nature that made him unique in all history. Only he was God himself present in a human manifestation.
In the next section, True Christianity #93, Swedenborg provides Bible quotes showing that the Holy One who came into the world is none other than the eternal God:
Since the angel Gabriel said to Mary, "The Holy One that will be born from you will be called the Son of God," I will now quote passages from the Word to show that the Lord in his human manifestation is called "the Holy One of Israel":
I was seeing in visions, behold a Wakeful and Holy One coming down from heaven. (Daniel 4:13, 23)
God will come from Teman, and the Holy One from Mount Paran. (Habakkuk 3:3)
I, Jehovah, am holy, the Creator of Israel, your Holy One. (Isaiah 43:15)
Thus said Jehovah, the Redeemer of Israel, its Holy One. (Isaiah 49:7)
I am Jehovah your God, the Holy One of Israel, your Savior. (Isaiah 43:1, 3)
As for our Redeemer, Jehovah Sabaoth is his name, the Holy One of Israel. (Isaiah 47:4)
Thus says Jehovah, your Redeemer, the Holy One of Israel. (Isaiah 43:14; 48:17)
Jehovah Sabaoth is his name, and your Redeemer, the Holy One of Israel. (Isaiah 54:5)
They challenged God and the Holy One of Israel. (Psalms 78:41)
They have abandoned Jehovah and provoked the Holy One of Israel. (Isaiah 1:4)
They said, "Cause the Holy One of Israel to cease from our faces. Therefore thus said the Holy One of Israel." (Isaiah 30:11–12)
Those who say, "His work should go quickly so we may see it; and the counsel of the Holy One of Israel should approach and come." (Isaiah 5:19)
In that day they will depend on Jehovah, on the Holy One of Israel, in truth. (Isaiah 10:20)
Shout and rejoice, daughter of Zion, because great in your midst is the Holy One of Israel. (Isaiah 12:6)
A saying of the God of Israel: "In that day his eyes will look toward the Holy One of Israel." (Isaiah 17:7)
The poor among people will rejoice in the Holy One of Israel. (Isaiah 29:19; 41:16)
The earth is full of guilt against the Holy One of Israel. (Jeremiah 50:29)
Also see Isaiah 55:5; 60:9; and elsewhere.
"The Holy One of Israel" means the Lord in his divine human manifestation, for the angel says to Mary, "The Holy One that will be born from you will be called the Son of God" (Luke 1:35).
From the passages just cited that describe Jehovah as the Holy One of Israel, you can see that although the names are different, "Jehovah" and "the Holy One of Israel" are one.
Many, many passages show that the Lord is called the God of Israel, such as Isaiah 17:6; 21:10, 17; 24:15; 29:23; Jeremiah 7:3; 9:15; 11:3; 13:12; 16:9; 19:3, 15; 23:2; 24:5; 25:15, 27; 29:4, 8, 21, 25; 30:2; 31:23; 32:14-15, 36; 33:4; 34:2, 13; 35:13, 17-19; 37:7; 38:17; 39:16; 42:9, 15, 18; 43:10; 44:2, 7, 11, 25; 48:1; 50:18; 51:33; Ezekiel 8:4; 9:3; 10:19-20; 11:22; 43:2; 44:2; Zephaniah 2:9; and Psalms 41:13; 59:5; 68:8.
This is only a small sample of the extensive scripture references Swedenborg provides in True Christianity and in The Lord to support his teachings about who Jesus Christ was. Space here will not permit a full representation of that scriptural support.
Rather than attempting to explain Jesus' process of glorification and provide supporting scripture references myself, I will offer this fairly lengthy quote from The Lord #35, which does both:
Step by step he took off the human nature he had taken on from his mother and put on a human nature from what was divine within him, which is the divine human nature and the Son of God.
It is generally known that the Lord was divine and human, divine because of Jehovah the Father and human because of the Virgin Mary. That is why he was God and a human being and therefore had a divine essence and a human outward nature, the divine essence from his Father and the human nature from his mother. This meant that he was equal to the Father with respect to his divinity, but less than the Father with respect to his humanity. It also meant that, as we are taught by the so-called Athanasian statement of faith, this human nature from his mother was not changed into or mixed with a divine essence, since a human nature cannot be changed into or mixed with a divine essence.
 All the same, this very statement of faith we have accepted says that the divine nature took on a human nature—that is, united itself with it as a soul with its body, so much so that they were not two but one person. It follows from this that he took off the human nature received from his mother, which was essentially like that of anyone else and therefore material, and put on a human nature from his Father, which was essentially like his divine nature and therefore substantial, thus making his human nature divine.
That is why the Lord is even called “Jehovah” and “God” in the prophetic books of the Word, and in the Word of the Gospels is called “Lord,” “God,” “Messiah” or “Christ,” and “the Son of God,” the one in whom we are to believe and by whom we are to be saved.
 Now, since from the beginning the Lord had a human nature from his mother and took this off step by step, while he was in this world he therefore experienced two states, one called the state of being brought low or being emptied out and one called the state of being glorified or united with the Divine called “the Father.” The state of being brought low occurred when and to the extent that he was primarily conscious of the human nature received from his mother, and the state of being glorified occurred when and to the extent that he was primarily conscious of the human nature received from his Father. In his state of being brought low he prayed to the Father as someone other than himself; while in his state of being glorified he talked with the Father as if talking with himself. In this latter state he said that the Father was in him and he in the Father and that the Father and he were one; while in his state of being brought low he bore trials, suffered on the cross, and prayed that the Father would not forsake him. This is because his divine nature could not be subject to any trial, let alone suffer on the cross.
These passages then show us that by means of his trials and the subsequent constant victories, and by means of his suffering on the cross, which was the final trial, he completely subdued the hells and completely glorified his human nature, as has been explained above.
 As for his taking off the human nature received from his mother and putting on the human nature received from what was divine within him called “the Father,” this we can see from the fact that whenever the Lord spoke directly to his mother he did not call her “mother” but “woman.” We find only three places in the Gospels where he speaks directly to his mother or about her, and in two of these he called her “woman,” while in one he did not acknowledge her as his mother. As for the two in which he called her “woman,” we read in John,
Jesus’ mother said to him, “They have no wine.” Jesus said to her, “What have I to do with you, woman? My hour has not yet come.” (John 2:4)
When Jesus from the cross saw his mother, and the disciple whom he loved standing by her, he said to his mother, “Woman, behold your son!” Then he said to the disciple, “Behold your mother!” (John 19:25–27)
The one occasion on which he did not acknowledge her is in Luke:
They announced to Jesus, “Your mother and your brothers are standing outside and want to see you.” Jesus answered and said to them, “My mother and my brothers are these who hear the Word of God and do it.” (Luke 8:20–21; Matthew 12:46–49; Mark 3:31–35)
In other passages Mary is called his mother, but never from his own mouth.
 There is further support for this in the fact that he did not acknowledge himself to be the son of David. In fact, we read in the Gospels,
Jesus asked the Pharisees, saying, “What is your view of the Christ? Whose son is he?” They said to him, “David’s.” He said to them, “So how is it that David, in the spirit, calls him his Lord when he says, ‘The Lord said to my Lord, “Sit at my right until I make your enemies a stool for your feet”’? So if David calls him ‘Lord,’ how is he his son?” And no one could answer him a word. (Matthew 22:41–46; Mark 12:35–37; Luke 20:41–44; Psalms 110:1)
We can see from all this that as far as his glorified human nature was concerned, the Lord was neither the son of Mary nor the son of David.
 He showed Peter, James, and John what his glorified human nature was like when he was transfigured before their eyes:
His face shone like the sun and his clothing was like light. And then a voice from a cloud said, “This is my beloved Son, in whom I am well pleased. Hear him.” (Matthew 17:1–8; Mark 9:2–8; Luke 9:28–36)
The Lord also looked to John “like the sun shining in its strength” (Revelation 1:16).
 We are assured that the Lord’s human nature was glorified by what it says about his glorification in the Gospels, such as the following from John:
The hour has come for the Son of Humanity to be glorified. He said, “Father, glorify your name.” A voice came from heaven, saying, “I both have glorified it and will glorify it again.” (John 12:23, 28)
It says “I both have glorified it and will glorify it again” because the Lord was glorified step by step. Again,
After Judas went out, Jesus said, “Now the Son of Humanity is glorified, and God is glorified in him. God will also glorify him in himself and glorify him immediately.” (John 13:31–32)
Jesus said, “Father, the hour has come. Glorify your Son, so that your Son may also glorify you.” (John 17:1, 5)
And in Luke,
Was it not necessary for the Christ to suffer this and enter into his glory? (Luke 24:26)
These things were said about his human nature.
 The Lord said, “God is glorified in him” and also “God will glorify him in himself” and “Glorify your Son, so that your Son may also glorify you.” The Lord said these things because the union was reciprocal, the divine nature with the human nature and the human nature with the divine. That is why he also said, “I am in the Father and the Father is in me” (John 14:10–11) and “All that is mine is yours, and all that is yours is mine” (John 17:10); so the union was full.
It is the same with any union. Unless it is reciprocal, it is not full. This is what the union of the Lord with us and of us with the Lord must be like, as he tells us in this passage in John:
On that day you will know that you are in me and I am in you. (John 14:20)
And in this passage:
Abide in me, and I [will abide] in you. Those who abide in me and in whom I abide bear much fruit. (John 15:4–5)
 Because the Lord’s human nature was glorified—that is, made divine—on the third day after his death he rose again with his whole body, which is not true of any human being, since we rise again with our spirit only and not with our body.
So that we should know this, and so that no one should doubt that the Lord rose again with his whole body, he not only said so through the angels who were in the tomb but also showed himself to the disciples in his human form with his body, saying to them when they thought they were seeing a spirit,
“See my hands and my feet—that it is I myself. Touch me and see, because a spirit does not have flesh and bones as you see I have.” And when he had said this, he showed them his hands and his feet. (Luke 24:39–40; John 20:20)
Jesus said to Thomas, “Reach your finger here, and look at my hands; and reach out your hand and put it into my side; and do not be unbelieving, but believing.” Then Thomas said, “My Lord and my God.” (John 20:27–28)
 To make it even clearer that he was not a spirit but a person, he said to the disciples,
“Have you any food here?” They gave him a piece of broiled fish and some honeycomb, and he took it and ate in their presence. (Luke 24:41–43)
Since his body was no longer material but had become divine substance, he came to the disciples when the doors were closed (John 20:19, 26) and disappeared after they had seen him (Luke 24:31).
Once the Lord was in this state, he was carried up and sat down at the right hand of God, for it says in Luke,
It happened that, while Jesus blessed his disciples, he was parted from them and carried up into heaven. (Luke 24:51)
and in Mark,
After he had spoken to them, he was carried up into heaven and sat down at the right hand of God. (Mark 16:19)
Sitting down at the right hand of God means gaining divine omnipotence.
 Since the Lord rose into heaven with his divine and human natures united into one and sat at the right hand of God (which means gaining omnipotence), it follows that his human substance or essence is now just like his divine substance or essence.
To think otherwise would be like thinking that his divine nature was raised into heaven and sits at the right hand of God, but not together with his human nature. This is contrary to Scripture and also contrary to the Christian teaching that in Christ God and a human being are like the soul and the body. To separate them is also contrary to sound reason.
It is this union of the Father with the Son, or of the divine nature with the human nature, that is meant in the following passages:
I came forth from the Father and have come into the world. Again, I leave the world and go to the Father. (John 16:28)
I go (or come) to the one who sent me. (John 7:33; 16:5, 16; 17:11, 13; 20:17)
What then if you were to see the Son of Humanity ascend where he was before? (John 6:62)
No one has ascended to heaven except the one who came down from heaven. (John 3:13)
Every one of us who is saved ascends to heaven, though not on our own, but rather through the Lord's power. Only the Lord ascended on his own.
Once again, this is only a small sample of Swedenborg's full explanation and biblical basis provided in True Christianity and The Lord. However, this should be enough to provide some idea of the answer to the question, "Who, according to Swedenborg, is this Jesus, born of a woman, who died and then was resurrected?"
What part did Jesus play in our salvation?
Now for a much briefer answer to the secondary question.
This is really a whole separate question of its own. There is not space here to provide a full explanation and biblical basis for it. Instead, I will offer Swedenborg's list of points on the subject of Redemption in True Christianity #114:
Redemption was actually a matter of gaining control of the hells, restructuring the heavens, and by so doing preparing for a new spiritual church.
Without this redemption no human being could have been saved and no angels could have continued to exist in their state of integrity.
The Lord therefore redeemed not only people but also angels.
Redemption was something only the Divine could bring about.
This true redemption could not have happened if God had not come in the flesh.
Suffering on the cross was the final trial the Lord underwent as the greatest prophet. It was a means of glorifying his human nature, that is, of uniting that nature to his Father's divine nature. It was not redemption.
Believing that the Lord's suffering on the cross was redemption itself is a fundamental error on the part of the church. That error, along with the error about three divine Persons from eternity, has ruined the whole church to the point that there is nothing spiritual left in it anymore.
For the full explanation and biblical basis for these points, please read True Christianity #114–133. The title link is to the book's listing on the publisher's website, where it can be purchased or downloaded for free. The numeral link is to a website where these sections can be read sequentially.
These are huge questions! Gaining proper answers to them requires considerable reading of the Bible and of Swedenborg's writings. However, this answer will, I hope, provide some taste and indication of Swedenborg's deep and satisfying answers to these excellent questions.