Will Adam and Eve be resurrected to eternal life or are they dead forever?
[Note : due to the number of scriptures quoted they are mostly shown as numbered footnotes.]
Firstly, there is a difference between immortality and eternal life. The soul is immortal and is not annihilated, despite the judgment of God on the first humanity : the return of the natural humanity to dust in the grave after death.
Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth? [Ecclesiastes 3:21 KJV]
Then shall the dust return to the earth as it was: and the spirit shall return unto God who gave it. [Ecclesiastes 12:7 KJV]
The coming of Jesus Christ, the Son of God, resolved the matter of immortal souls in a disembodied state and their bodies returned to dust. (1)
Firstly, through faith (and by remission of sins and justification by faith) those who believe on the name of Jesus Christ receive the Holy Spirit (2) are born of the Spirit (3) they are born of God (4) and, receiving the Person of the Holy Spirit into union with their spirit, they receive the gift of eternal life : divine life which is eternal, in union with their own created spirit.
Secondly, by resurrection (or, if still on alive on earth at the coming of the Lord, immediately) they receive, through the redemption which is in Christ Jesus (5), a glorious body like unto Christ's body (6).
Thus 'eternal life' is much more than just 'prolonged mortality' or 'immortality'.
Those who have not believed on Jesus Christ are condemned already for their unbelief and shall not partake of that glorious resurrection (7). They shall be resurrected to damnation and their place shall be in a lake of fire : the eternal destruction of their immortal souls where their worm dieth not and the fire is not quenched (8).
Who shall be punished with everlasting destruction [2 Thessalonians 1:9 KJV]
As to Adam and Eve there is good cause to believe that they were saved from their transgression.
Adam accepted the judgment of God and, once informed of a seed to come (of woman but not of man) that would bruise the serpent (9) from above (that is to say, humanity ascended above created spirit) he named his wife appropriately (having not done so, before) (10).
Once Adam had done this - a step of faith - God responded and clothed both Adam, himself, and his wife in coats of skins (11) denoting a sacrifice and denoting that they were 'clothed' with the skin of another : surely a sign of redemption. That is to say a sign, immediately, of eventual redemption when the humanity, through whom that redemption would come, would be manifested and raised in ascension, above the created spirit with whom they had conspired.
More than this, when Cain did not take responsibility (my 'avon' is greater than I can bear) for the death of Abel (12) he chose to 'go out from the presence of the Lord' (13). Practically, what he did was to leave the household of Adam and Eve.
Now, Adam and Eve were banished from Eden, wherein they had been in God's presence. But now, banished from Eden, they are still in 'the presence of the Lord'. This would point to them being an household of faith. And, because of faith, the Lord is with them. Although God's judgment is upon the first humanity, as such, and that humanity is forever banished with a sword protecting the tree of life, yet those who believe in a sacrifice yet to be made, are in God's presence by faith.
And when God speaks to Cain in judgment, it seems quite reasonable to suppose that the words actually came from the mouth of Adam. Inspired of God, but spoken of Adam.
Thus I would suggest that Adam obtained faith, despite his transgression. And I would suggest that Adam shall, in due course be raised from the dead in a glorified body, since he is (now) a partaker of eternal life through faith.
Though all was not yet revealed, yet what was revealed (at the time) Adam believed. And of Adam (I would suggest) it is as it was with Abraham :
Abraham believed God and it was accounted to him for righteousness [Genesis 15:6 and Romans 4:3 KJV]
(1) Now if Christ is preached that He rose from the dead, how do some among you say that there is no resurrection of the dead? [I Corinthians 15:16 KJV]
(2)Received ye the Spirit by the works of the law, or by the hearing of faith? [Galatians 3:2 KJV]
(3) ... that which is born of the Spirit, is spirit [John 3:6 KJV]
(4) Whosoever is born of God does not practice sin [I John 3:9 KJV]
(5) Being justified freely by his grace through the redemption that is in Christ Jesus: [Romans 3:24 KJV]
(6) Who shall change our vile body, that it may be fashioned like unto his glorious body [Philippians 3:21 KJV]
(7) He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him. [John 3:36 KJV]
(8)Where their worm dieth not, and the fire is not quenched. [Mark 9:44 KJV]
(9)And I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel. [Genesis 3:15 KJV]
(10)And Adam called his wife's name Eve; because she was the mother of all living. [Genesis 3:20 KJV]
(11) Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them. [Genesis 3:21 KJV]
(12)And Cain said unto the LORD, My punishment is greater than I can bear. [Genesis 4:13 KJV]
(13)And Cain went out from the presence of the LORD, and dwelt in the land of Nod, on the East of Eden. [Genesis 4:16 KJV]