Do Protestants understand Jesus to be both "the son of Adam" and "the son of God"?
Protestants believe that Mary was a virgin when she conceived by the power of the Holy Spirit, but they do not go along with the Catholic view that Mary herself was without sin. The Bible says nothing about Mary being sinless.
The Protestant view is that Jesus was fully human and this came from Mary being his biological mother. Jesus alone was born of a virgin, conceived by the Holy Spirit. Jesus had no human father. The fact that Jesus was without sin is clearly stated in the Bible:
For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God (2 Corinthians 5:21).
You know that he appeared in order to take away sins, and in him there is no sin (1 John 3:5).
I am unfamiliar with the expression that Jesus was “in Adam” but the Bible likens Jesus to the first man Adam:
The first man Adam became a living being; the last Adam, a life-giving spirit. The spiritual did not come first, but the natural, and after that the spiritual. The first man was of the dust of the earth, the second man from heaven. As was the earthly man, so are those who are of the earth; and as is the man from heaven, so also are those who are of heaven. And just as we have borne the likeness of the earthly man, so shall we bear the likeness of the man from heaven (1 Corinthians 15:45-49).
Just as Adam was the first of the human race, so Christ is the first of those who will be raised from the dead to eternal life. Adam came first on this earth and was made from the dust of the earth. While it is true that Christ has existed from eternity past, He is here called the second man or second Adam because He came from heaven to earth many years after Adam. Christ came as a human baby with a body like all other humans, but He did not originate from the dust of the earth as had Adam. He “came from heaven.” Source: https://www.gotquestions.org/Jesus-second-Adam.html
That, as far as I am aware, is the Protestant view of Jesus being both fully human and fully God. Mary was a descendant of Adam and, like all humans, was herself subjected to inherited sin. Jesus was without sin because he came from heaven, from above, and no human father was involved.
Additional information regarding the Baptist view of Jesus being without sin:
The Baptist Confession of Faith (1689) Section 8: Christ the Mediator The Son of God, the second person in the Holy Trinity, being true and eternal God, the brightness of the Father's glory, of the same substance and equal with Him;
Who made the world, and who upholds and governs all things which He has made,
did, when the fullness of time had come, take upon Himself man's nature, with all its essential properties and common infirmities, with the exception of sin.
He was conceived by the Holy Spirit in the womb of the Virgin Mary, the Holy Spirit coming down upon her and the power of the Most High overshadowing her, so that He was born to a woman from the tribe of Judah, a descendant of Abraham and David, in accordance with the Scriptures.
Thus two whole, perfect and distinct natures were inseparably joined together in one person, without conversion, composition, or confusion;
So that the Lord Jesus Christ is truly God and truly man, yet He is one Christ, the only Mediator between God and man. http://www.romans45.org/creeds/bcof.htm#part8
The New Testament writers had to combat a form of Gnostic teaching that denied the full humanity of Christ. They believed flesh was evil but spirit was pure, therefore God would never have incarnated as a fleshly human. They said Christ only appeared to be human, but it was an illusion – he was actually spirit. The New Testament declares Jesus was human and also God incarnate.
Just to be clear, I am answering the question :
Do Protestants understand Jesus to be both "the son of Adam" and "the son of God"?
Also, it is to be noted that Jesus called himself 'son of man' not 'son of Adam', the terms having very different connotations.
Martin Luther says in his Disputation on the Divinity and Humanity of Christ :
May you preserve this article in its simplicity, that in Christ there is a divine and a human nature, and these two natures in one person, so that they are joined together like no other thing, and yet so that the humanity is not divinity, nor the divinity humanity, because that distinction in no way hinders but rather confirms the union! [Preface]
Here, Luther is expressing that the union of humanity and Deity is not a merging of natures but a union of natures in the Person. Only in the Person of Christ is this union effected. As to the human nature of Jesus Christ, that was formed within the virgin, Mary, without any introduction of anything from Joseph.
Neither substance nor sin was contributed by Joseph to the forming of the humanity within Mary.
John Calvin says in his Institutes of the Christian Religion (Chapter 12 Section 3):
Therefore, our Lord came forth very man, ... that he might in his stead obey the Father; that he might present our [sic - I presume he means 'his'] flesh as the price of satisfaction to the just judgment of God, and in the same flesh pay the penalty which we had incurred.
Moreover, the expected Redeemer was that son of Abraham and David whom God had promised
I would point out that Luther uses the term 'true man' and Calvin uses the term 'very man' of Christ. Neither (that I can find) use the term 'fully man', which implies that there is some kind of 'not-fully-man' which might exist.
Calvin here is careful to express 'son of Abraham' and 'son of David' but not 'son of Adam'.
The genealogies in Matthew and Luke both demonstrate the rights of Jesus Christ (in Matthew his royal rights and in Luke his human rights) but it is patently clear that those rights are not by origin from Adam but by adoption by Joseph.
Mary was of the tribe of Levi, for we are told that Elizabeth, of the daughters of Aaron, was a close relative of Mary. Therefore Jesus inherited the royal rights by adoption from Joseph (who was of the tribe of Judah) who passed on his place as a son of David to Jesus, his adopted (but not biological) son.
In Luke's genealogy the wording is careful stating that Jesus 'being (as was supposed) the son of Joseph which was of Heli which was of Matthat ...
Thus there is no personal connection between Jesus and Joseph other than the fact of adoption. And no personal connection of Jesus to Heli (or any of the others in the genealogy) since it was Joseph who was 'of' Heli.
The Archangel Gabriel's words, Luke 1:26-38, are very distinct indeed in the original Greek, regarding the begetting within Mary. And they are very distinct indeed regarding that which was 'holy' taken from Mary.
The Apostle John asserts that Jesus Christ is 'come in the flesh' I John 4:2 and the writer to the Hebrews declares that :
... in all things it behoved him to be made like unto his brethren ... he took on the seed of Abraham. Hebrews 2:17 and 2:16.
'Took on the seed of Abraham' is very specific and the original is 'take hold' the word being the same as that used of conception. The taking hold of seed is within the virgin, it is not a matter of personal descent.
Jesus Christ is come in the flesh, but is untainted by the sin which descends from Adam by natural generation. Never is he called 'the son of Adam'.
The promise made in Eden that the seed of the woman shall bruise the head of the serpent has been fulfilled. It is the seed of the woman by whom God's purpose in creation has been realised, not the seed of the man, Adam. Woman taken out of Adam before he sinned, is the one through whom came the fulfilment of all God's purposes in Creation.
As we see in Revelation 12 - a woman clothed with the sun and the moon under her feet is the one who brings forth the man-child. The man-child (the Greek word arrhen means 'lone man' or 'solitary man') has no father on earth.
His only father is God, the eternal, and he is caught up to heaven immediately. So saith Jesus as he stood on human feet upon earth :
And no man hath ascended up to heaven, but he that came down from heaven, even the Son of man which is in heaven. [John 3:13, KJV.]
The concept of being FULLY HUMAN and FULLY GOD isn’t really that hard to understand as long as we remember that;
John 4:24 KJV God is a Spirit: and they that worship him must worship him in spirit and in truth.
Spirits have no material characteristics. We have no problem believing that we have a physical body and a Soul. The soul has no material characteristics just as God has no material characteristics. Also we as Protestants are well aware that we are human and descended from Adam. We understand that human bodies are separate from our souls and yet our fully human bodies and our souls abide together even though both are separate parts of our being. We also fully understand that our minds contain knowledge of both good and evil; no thanks to Adam and Eve. What we have the hardest time understanding is that Jesus had the mind of God. So what does having the mind of God actually mean?
Genesis 2:9 KJV And out of the ground made the LORD God to grow every tree that is pleasant to the sight, and good for food; the tree of life also in the midst of the garden, and the tree of knowledge of good and evil.
In the above Scripture we find that God knew of good and evil before even making mankind. Knowing good and evil is not the same as sin. God did not say to Adam that if he gained the knowledge of good and evil he would die, but instead said:
Genesis KJV 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
God in his foreknowledge knew that Adam would willfully disobey since Adam was not fooled.
So therefore if Jesus knew both good and evil; yet did not disobey God he would remain sinless. So we must know if Jesus being fully human was without original sin. This can be established only if we know that Jesus did have all knowledge and yet was totally obedient to God, and was not guilty of the sin that Adam committed.
Luke 10:18 KJV And he said unto them, I beheld Satan as lightning fall from heaven.
In the Scripture above Jesus tells us he was present when Satan was ejected from Heaven, He could only have done that if his mind were present on that occasion. Since it is gained knowledge, or as in this instance a part of Omniscience. Since Omniscience is of God then God must be present in Jesus, and since God is Spirit, then Jesus must have had the spirit of God and not the spirit of man, and not the soul of mankind since the Soul is also Spirit.
This is just a quick look at a few of the pertinent Scriptures, and must be accompanied by more study in order to fully comprehend the concept of the God man inter connection of Jesus. Much of this relationship is scattered throughout the entire Bible.