There are many big questions here. It will not be possible to give a full answer to all of them in a single answer. But I can provide at least a general answer to most of them.
Why did Christ die?
Christ's death did not "pay for our sins" (something the Bible never says), nor was it in itself our salvation. Rather, his crucifixion was the occasion of his final battle against the power of the Devil, which is a personification of hell: the combined power of all human evil.
We are given several vignettes of Jesus' temptation battles against the Devil, the primary ones being his temptation by the Devil after his baptism and his forty-day fast in the desert, and his temptation in the garden of Gethsemane before his crucifixion. He also engaged in verbal battles with the corrupt religious leaders of the day, which were an earthly arm of the spiritual evil and corruption represented by the Devil.
On the Cross, Jesus fought his last battle against the Devil, and achieved complete victory over the power of evil through his resurrection and subsequent ascension to the throne of God. For one of Swedenborg's explanations of this, including supporting passages from the Bible, see Doctrine of the Lord #12–14.
Did Christ die for our sins?
I am extracting this shortened version of the question because it is important to know what it means to "die for our sins." This, once again, does not mean that Jesus died to pay the penalty for our sins. The Bible simply never says this. It also does not mean that Jesus died instead of us. Nor does it mean that Jesus died to appease the wrath or satisfy the justice of the Father. Once again, the Bible simply never says this.
Jesus died for our sins in at least two senses.
First, it was our sins that killed Jesus. God did not kill Jesus. The Jewish religious authorities handed him over to the Roman governor Pilate, who ordered him crucified. In other words, it was evil and sinful humans, not God, who murdered Jesus. He died due to our sins.
Second, Jesus died to protect and save us from human evil and sin.
As an analogy, picture a frontiersman who lives in the woods with his wife and children. Outlaws are approaching his house to break in, steal his belongings, rape his wife, and kill his children. To protect his wife, family, and home, the man goes out with loaded guns, shoots the outlaws, killing some of them and wounding others, chasing them off. However, he is fatally wounded by their return fire.
In the same way, Jesus stood between us and the power of human evil, personified in the Bible by the Devil and Satan, saving us from death and destruction at the hands of the Devil, but dying in the process. (However, he then rose back to life.) If he had not done this for us, the Devil would have broken in and spiritually raped and murdered us, dragging us down to eternal death in hell.
Did Christ die for our sins as an offering as we find in the Old Testament?
Due to the false and unbiblical Catholic satisfaction theory of atonement and its Protestant penal substitution variant, Western Christians have badly misunderstood the nature of sacrifices in the Old Testament.
Despite the faulty translation of Leviticus in the New International Version, the New Revised Standard Version, and some other modern Bible translations, the offerings of the Old Testament were not "penalties for sin." This is a complete misunderstanding of how the sacrifices functioned in ancient Hebrew culture.
Rather, they were feasts with God that symbolized and brought about a rapprochement (or "atonement") between humans and God. Though some animals sacrificed were burnt whole upon the altar, for most sacrifices only part of the animal was burnt on the altar, another part was eaten by the priests and their families, and the rest was eaten by the person bringing the sacrifice and his family and friends. For the people offering the sacrifice, it was a literal feast and a joyful occasion, not some "payment for sin."
The parts burnt on the altar were God's part of the feast: "a sweet savor unto the Lord" (Leviticus 1:9, 13, 17, etc.). The parts that the priests and their family ate signified the sacred nature of the sacrifice. And the parts that the person offering the sacrifice and his family and friends ate represented their acceptance of the Lord's presence in their lives. The function these sacrificial feasts was similar to the function of feasts that kings of two different nations, and their nobles, had with one another to seal and celebrate a peace treaty between their two kingdoms.
This was how sacrifices functioned in ancient Israelite society. It had nothing to do with paying for sins, as if the person offering the sacrifice were forfeiting a valuable animal as a punishment for sin. No. The sacrifices brought together God, the priests as God's representatives, and the people offering the sacrifice for a shared meal that represented the people accepting God's presence, including committing themselves not to sin, but to live according to God's commandments.
In short, the sacrifices represented the renewal of the relationship between God and God's people.
This is also the meaning of the New Testament metaphor of Christ as a sacrifice for our sins. It was not a payment to the Father. The Bible simply never says this! Rather, it was Christ offering us a sacred meal with himself, "God With Us" (Matthew 1:23), so that we could be reconciled to God and live in harmony and communion with God.
This becomes very clear from the fact that Christ replaced all of the Jewish sacrifices with a meal shared with himself. That meal is the Holy Supper or Eucharist that Christians continue to celebrate to this day. It represents our living relationship with the Lord God Jesus Christ.
Does Jesus' blood cover sin?
On the meaning of "covering sin" in the Bible, please see this related question and answer here on Christianity StackExchange:
How did Swedenborg interpret 1 John 2:2: "He is the propitiation for our sins"?
I will not repeat everything in my answer to that question, but here is the short version: The "covering of sins" is the Lord's merciful forgiveness of our sins. We accept that forgiveness when we repent from our sins and begin living a new life through the power of the Lord working within us.
It is also necessary to understand that the blood of Jesus is not literal blood. In John 6:53–56 Jesus proclaims to the crowds who had come to listen to him:
Very truly, I tell you, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Those who eat my flesh and drink my blood have eternal life, and I will raise them up on the last day, for my flesh is true food, and my blood is true drink. Those who eat my flesh and drink my blood abide in me and I in them.
And yet, when he celebrated the Last Supper with his disciples, he did not give them his flesh to eat and his blood to drink. Rather, he gave them bread to represent his flesh, and wine to represent his blood. This should make it very clear that it is not Christ's literal blood that covers our sins.
For a fuller presentation of this, please see this article on my blog:
Eat My Flesh, Drink My Blood
There is not space here to fully present and support from the Bible the meaning of the blood of the Old Testament sacrifices, and of Christ's blood in the New Testament. For one of Swedenborg's explanations of the meaning of the blood of the sacrifices, see Arcana Coelestia #9393
Here, once again, is the short version:
If it is not Christ's literal blood that covers our sins and cleanses us from them, then it must be Christ's spiritual blood that does this. And Christ's spiritual blood is none other than the spiritual truth that the Lord teaches us in the Bible. If we live according to that truth, then our sins are "covered over" in the sense explained in the Christianity StackExchange question and answer on "propitiation" linked just above, and they are also forgotten and done away with because we are no longer committing them, but are living a good life instead. For a different version of this based on the prophet Ezekiel in the Old Testament, please see this article on my blog:
Ezekiel 18: God’s Message of Hope . . . If You Think there’s No Hope for You
All of this is represented and symbolized by our participation in the Holy Supper, or Eucharist. When we eat the bread, we are accepting God's love into our life, which is the spiritual significance of bread, and also of the flesh of the Old Testament sacrifices. When we drink the wine, we are accepting God's truth into our lives, which means living according to the teachings and commandments of the Lord.
This is the true spiritual feast with God that is represented symbolically by the sacrifices of the Old Testament, as suggested in the spiritual interpretation of the sacrifices presented in the Epistle to the Hebrews in the New Testament.
As mentioned in the Introduction, these are big questions. This brief answer in no way does them justice. The linked material provides more detail and biblical support for some parts of this answer.
I hope this much offers some of the required pieces and gives some of the clarity that you are seeking. If you want more detail on any one of your questions, please do ask it as a separate question here, and I will do my best to respond.