I'll focus on the experiments.
Let's for the sake of experimentation hypothesize that reincarnation is real and that each separate human body gets a separate soul that contains personality characteristics, as well as whatever else a soul has.
The we can do the well known identical twin studies, with a twist.
The usual identical twin study investigates the role of the environment in human development versus genes. It's the nature or nurture question. But this question/answer assumes that environment and genes are the only influencing factors.
If there is truth to reincarnation, and if each of the identical twins gets a soul with a different history of past lives, a reasonably assumption, not knowing how these things are decided, then each of the identical twins should have a distinct personality.
So we need to examine identical twins that have identical nurturing as well as natures.
Now agreed that identical nuturing is hard to come by. Even in the womb, one twin is dominant. So the scientists that do these studies will have to bias out the wellknown identical twin variations.
The difficulty now is to decide what to actually test that would be expected to be unaffected by dominance and similar effects. To be truthful I'm a little at a loss here, even though I ran track with a set of identical twins who had differing personalities. I could not tell them apart from loking at them, or even from their running. But after talking with either, despite exactly the same voice, I could sense which twin I was talking to.
On the other hand, perhaps the soul enters the womb at conception. Then presumably this would be nature's way to multiple souls- to produce two souls from one. But if the soul enters some days after conception, as the occult liyterature says, then we would expect somewhat distinct personalities. Do you know any identical twins that were bought up together?