Does Neanderthal exist earlier than Adam & Eve?
Science says they are not homo sapien.
Do they have soul? Made in the image of God?
That depends on how one reconciles Adam and Eve with the hominid family tree; or whether they try to reconcile them at all. As some Christians will straight up deny the existence of a hominid family tree altogether; and some Christians might not view Adam and Eve as literal-historical people; representing two polarizing view points. Most Christians, however, are going to be somewhere in the middle of those two view points--but how they approach the subject may vary quite a bit.
If Adam and Eve are ancestral to all or most of all in the genus Homo, then no, neanderthals wouldn't pre-date Adam and Eve. If Adam and Eve are only ancestral to Homo sapiens, then possibly yes--depending on whether neanderthals are a sub-species of Homo sapiens (Homo sapiens neanderthalis) or a sister-species to Homo sapiens (Homo neanderthalis). Modern humans and neanderthals were close enough that they could produce viable offspring--many people today have neanderthal DNA. Which means either a sub-species or a very closely related sister-species.
Those who believe in a Young Earth Creationist view would likely argue that neanderthals are just another group of humans, not a different sub-species or species at all, since there would be a rejection of human evolution altogether. In which case, again, Adam and Eve would be ancestral to them.
The best answer, I think, is that there's no way to actually say anything positively at all. All answers are going to be guesses, not actually answers.
I don't know how neanderthals relate to Homo sapiens, and I don't know how Adam and Eve relate to neanderthals or other hominids.
But I do believe that God alone is the Giver of life from whom all living things have the breath of life, the Maker of heaven and earth, of all things seen and unseen. And I believe that it is from God alone that we have our rational soul, our spiritual life, our reason, and that we bear the Divine Image because God gave us and made us in His image. And I believe that this Divine Image has been injured, marred, and broken by the Fall, through the disobedience of Adam sin and death have become a plague, and we all suffer from it. We are born into this world with the guilt of sin hanging over us and we need the Gospel of Jesus Christ and the salvation and freedom that comes from Him alone who undoes and breaks Adam's curse for us.
How everything fits together I don't know. But neither does it matter for confessing the faith:
I believe in one God, the Father Almighty, Maker of heaven and earth. And in one Lord Jesus Christ, His Son. He was conceived by the Holy Spirit and born of the Virgin Mary, He suffered under Pontius Pilate, was crucified, died, and was burried. He descended into hell, He rose on the third day, He ascended into heaven and sits at the right hand of the Father, from whence He will come again in glory to judge the living and the dead. I believe in the Holy Spirit, the holy catholic Church, the forgiveness of sins, the resurrection of the body, and the life everlasting. Amen.
Glory to the Father, and to the Son, and to the Holy Spirit, as it was in the beginning, is now, and ever shall be. Amen.
-CryptoLutheran