I like your post and the points you raise. My practice is based on the 7 factors of Awakening. Investigation is a very important part of the path for me and it seems to me you are doing that quite nicely.
I will offer some of my 'ideas/insight' on the rebirth and reincarnation.
The basic premise is that we are reborn into the world after we die. In other words we continue to be a part of the world after we die. So our self continues to exist in the physical world after our biological body has died. But hold on... doesn't Buddhism also teach that there is no self.
So something about us that is not self, is reborn into this world after we die. Some suggest there are realms other than the earth where we could be reborn.
Lets go back to original question regarding... Something about us that is not self, is reborn into this world after we die.
But how can this make any sense to a rational mind? How do we investigate something like rebirth and as you suggest seek proof?
Meditative perceptions are very important to the understanding a Buddhist has about our existence. You may be interested in the following book.
This is a lived philosophy in the deepest sense possible, in which theoretical positions are to be ingrained in the mind as the result of diligent practice. We are not speaking of ideas but of intense, dedicated, meditative practice. The philosophical doctrines of Buddhism thus emerged from specific, well-defined meditative perceptions, which were part of a sustained effort to give new shape to (or at times possibly to transcend) experience. The early Buddhist teachings were first of all verbalized reflections on meditative events.
Shulman, Eviatar. Rethinking the Buddha: Early Buddhist Philosophy as Meditative Perception (p. x). Cambridge University Press. Kindle Edition.
Having said that, based on my own meditative perceptions and my years of practice and investigation, I have come to see rebirth as a inherent aspect of our biological world. Many will say this is the wrong view but it is regardless how I view things.