First, consider that “What we have always understood” is not always what the Bible actually teaches. let’s start with the very concept of “human” v “animal”. The Bible, with the possible exception of Genesis 1 under GAE, is always using the category of adam, ie the decendants of Adam - and so will not be giving positive teaching on the nature of anyone outside the garden, whether “non-Adamic Homo sapiens” under GAE or, come to that, Neanderthals or Homo erectus under any other scheme trying to define “human” biblically.
So, regarding your three criteria, I would question whether the Bible actually teaches such a threefold definition.
Certainly, the creation of man is “in the image and likeness of God,” but as Josh has stated already, that concept is worthy of much discussion (which he does extensively in his book, and I from a slightly different angle in mine). In Gen 2 Adam was created outside the garden (presumably, but not overtly stated) in God’s image. But his unique status arose only from his being introduced into the garden of God. He acquired thereby a special relationship with God, and only in that context does the idea of a fall, or of salvation, have any real meaning.
Genesis also relates eternal life not to Adam’s creation, but to his presence in the garden with the tree of life. As Jordan suggests, there are significant problems with the concept of “eternal soul,” as understood by philosophically-trained people like Gregory or Thomas Aquinas, but even more so now, when we are so influenced by Descartes’ concept of the soul as a “ghost in the machine” rather than “our animating principle.” The OT concept of “soul” (nephesh) is, usually, simply the concept of “living,” and not a technical term at all.
Then freedom of the will. Is free will possible without sin? Well, our faith stems from Jesus, who had the freest of wills, but always obeyed his Father. And the hope of the Kingdom, if I’m not mistaken, is that we shall be like him, freely righteous and in that sense incapable of sin. If the modern idea that freedom inevitably leads to sin is true, then the hope of eternal righteousness is either false or predicated on eternal slavery, which is definitely not the Bible’s teaching!
But I want to confirm Joshua’s point that “if they had never been given a command from God, they could never have transgressed that command.” He did not pluck this out of the air, but from Romans 5:13, which in context is teaching that what constitutes sin, especially the fall of Adam, is not primarily anything about moral consciousness, but about disobeying the covenant command of God. Without some inherent ability to choose, Adam could obviously not have transgressed, but equally without a personal relationship of command/compliance, sin in the biblical sense cannot be imputed.
I would add that, being sinners in a fallen race, we often fail to appreciate how that first sin led (as described in Genesis itself) to an escalation of perverted morality. This wilful distancing from God (cf Rom 1:28-32) would not apply to any people outside the garden. My own conclusion (from various evidences) is that such people would have followed their created nature in a “natural” religious sense, but lacking the intimate relationshipo with God that was the purpose of Adam’s call; amongst other things, he failed mankind by not making that relationship generally available, instead progressively implicating them in his sin and necessitating the grand plan of salvation that begins in Gen 12 and culminates in Christ.
So could they be “saved”? Saved from what, and for what? They were not under God’s judgement… but neither had they been given any hankering after a life over and above their created nature.
One final note in response to Allen on Gregory’s anthropology: I would assume that his invocation of the vegetative, animal and human soul does not imply three souls, any more than it does in Aquinas. They are more like progressive stages of “animation,” in that plants have vegetative functions, animals retain those but also have mobility, volition etc, and human beings have all that, but rationality too. The word that’s been coined in modern times for that basically Aristotelian concept is “hylemorphism”, basically meaning the “form” (morphe) that matter (hyles) takes on to do what it does.
Aristotle reasoned to an eternal sould from the idea that “reason” is immaterial and therefore imperishable - which is not entirely watertight as an argument. He was also having to deal with the biblical concept that, even prior to the final physical resurrection, the righteouss dead are kept safe in Christ (and the wicked for final judgement). But as far as I can see, the Bible leaves the “manners and means” of that pretty vague: it is not teaching the intriniscally eternal nature of the soul.