This is the problem of “first beginnings”, which the Buddha said is unknowable. See SN 15.1, etc.
Not getting into the tarpit of philosophy/theology on the “prime mover” is one of the most canny and powerful moves that the Buddha made. What matters is that suffering is real, and we can do something about it.
Adding an assumed “unconditioned” is a non-explanation. It’s like saying “God did it”. It’s just a word, a word that evokes a response, but which doesn’t really mean anything.
How could the unconditioned condition anything? Christian theologists have dug themselves into endless pits trying to explain how God, who is unconditioned, created a world that is conditioned. We don’t have that problem, thank goodness!
These are two separate questions. Memories of past lives are conditioned, an aspect of the larger process of the unfolding stream of consciousness. Knowing what a previous state of consciousness was does not require knowing what an assumed original state was.
Let’s say I wanted to figure out the route taken by a particular car, so I can see where it’s headed. If I see it coming down 4th Street, I can infer that it came from 3rd Street. And if I see its indicator on, I can infer that it’s turning right on 5th Street. It’s 7pm, and I know that 5th Street leads to a popular restaurant, so I can guess that maybe that’s where it’s heading.
But say I was to add to my information the place where the car was manufactured. If it was built in Korea or Japan or Germany, does that help me at all? It does not.
And what if it is my car, and I am sick of it, and want to step out of it and leave it behind: would it matter where it was made?
Knowledge of the ultimate origins of consciousness is a fascinating question, but it is futile. We will never know the answer, so speculation is merely a distraction from what is really relevant.
The question of how the process of rebirth occurs is explained in the suttas through dependent origination. It is a continual unfolding of conditions. By seeing how consciousness flows and changes in the present, we can understand how that works in the past and future. This is what’s important for spiritual progress.
In the various Abhidharma schools, refinements and elaborations of this theory were developed in various different ways. If someone wants to do a modern take on it in terms that a scientist could grapple with, then great. But it won’t change the path.