An Abridged Version of your Original Question is:
Is reincarnation inevitable?
The Short Answer
If reincarnation is inevitable, it is not for the reasons you provided. In your description, you assume that your future physical self would be the same as your current self. However, reincarnation necessitates a different body. Maybe you would be reincarnated as a tuna fish. Even if you were re-incarnated as a person, it would be a different person.
The Long Answer
After reading the details of your question, it seems like you have re-invented something called The Infinite Monkey Theorem. A lot of people have had the same idea throughout history. If you click on the link I provided, then you can read at least the first paragraph or two of the Wikipedia page written about the theorem.
When I was in college, I wrote a few programs which randomly generated "test data." For example, I had an assignment to create a computer program for managing a contact's list: The computer program (app) would keep track of phone numbers and email addresses of your family members, your boss at work, etc... I wrote a program which randomly generated people's first names, phone numbers, etc... I occasionally, I saw short English words, such as "dog" or "cat" appear in the street names. If you randomly generate a sufficiently large amount of data, short English words will appear.
Suppose that you had a machine which randomly generated letters, punctuation marks, and other symbols at a rate of 80 words per minute.
A person who can type at a rate of 80 words per minute is not unheard of. Computer can go faster than that, but I choose that speed so that you can imagine checking in on the machine from time to time. You can check and what the machine is either displaying on a computer screen, or printing on paper.
This machine outputs exactly one character at a time, such as "A" or a comma, or a space.
If the machine magically lasted forever, never broke, and printed letters for all eternity, then what would happen? Well, the machine would eventually type the question, "Is reincarnation inevitable?" Note that the machine did not "learn" about the English language. The machine is not an artificial intelligence (AI). If you wait long enough, and the letters are typed are random, the machine will eventually accidentally type "reincarnation." You might have to wait 1 billion years, but it will eventually happen.
When I was in elementary school, I once asked my father if time was infinite, and you waited long enough, would asteroids in space randomly smash together to build a Ford Pick-up Truck. I did not mean that a truck would magically appear in a fraction of a second, but rather, over millions of years, a piece of copper there, and a piece of iron here, would eventually be welded onto a partial-truck by asteroids hitting it. Most work-in-process trucks would be destroyed before completion. Clearly, rocks randomly smashing together to form a pick-up truck is unlikely, but if you waited long enough, would it happen? My father said no. However, mathematicians have proved the infinite monkey theorem.
As a reminder, your original question was:
Is reincarnation inevitable?
I read the details. I think that you were trying to imagine a case where:
- The universe never ends?
- The events in the universe is sufficiently chaotic
If that were true, then someday, there will be a planet like earth again. Also, there will eventually be a planet like earth which has humans living on it. A person very much like you will be born again.
An idea very like that was expressed in the last 10 minutes of the Hollywood movie K-PAX (2001). If you have not watched X-PAX before, then I recommend watching the movie for entertainment purposes; but it the movie does not contain any secret truths.
I am not sure if the universe and time and stuff will go on and on, forever and ever. However, that is okay, because it is clear that even if time never ended, etc.. etc... what you described would not be reincarnation.
That is, if there are a billion billion big bangs, and you pop into existence again a long time from now, that person will not be a reincarnation of you.
Part of the definition of "reincarnation" is that your future self will not have the same body as your current self. For example, in the future, you might be a squirrel instead of your current state as a human.
Not only that, but the idea of reincarnation includes a lot of stuff about souls and the supernatural. If a billion billion years from now, there is another human which is physically just like you, I am not sure that shows that souls exist, or magic exists, or that the future body with your same genetics would contain the same soul, and not a new soul.
The infinite monkey theorem does not prove that reincarnation is inevitable. This is because a copy of your body, or a copy of the collected works of shakes-spear, or whatever, does not have your same soul.
I personally do not believe in the existence of souls or reincarnation (for much the same reasons) but I do understand the basics of the ideas. Even if it were possible for a person to randomly be born someday with your genetics, and other physical things were the same too, that would not reincarnation. In fact, reincarnation discusses having new/different bodies, but you assume that your future self would have the same body.
Someone could reply, "as unlikely as it is, given an infinite amount of time, I will eventually be re-born as a fish."
Well, the problem is that, from a scientific standpoint, the fish would not be you.
If you allow yourself to have a fish body, but keep the same soul, then it no longer makes sense to talk about probability theory, the infinite monkey theorem, or accidentally re-typing Shakespeare's collected works. If magic exists, then you can do anything. Scientific rules about probability, and future events, do not matter anymore if you allow supernatural events.
The answer is that things you described are not sufficient to conclude that reincarnation inevitable.
You are essentially arguing that "as unlikely as it is, given an infinite amount of time, I will eventually be re-born" is a corollary of the infinite monkey theorem.