First, I don't remember why we're talking about the Mirror Universe in a thread about timelines, and if I'm the one who brought it up, I'm sorry.
My point was this: In alternate timelines, as depicted in "Star Trek" and elsewhere, each reality tends to have the same basic laws of science -- atoms, electrons, gravity, biology, genetics, thermodynamics, entropy, etc. That means, even in an "infinite" number of alternate timelines, NONE will have a square-shaped Earth instead of a round one, because all realities will have gravity. (If you want to argue that there are realities without gravity, then we will have to redefine the word "reality.")
In fact, super string M-theory says there are universes out there, where indeed there is no gravity, where space can be a fluid (heh, heh), where there is no weak nuclear force, or the force acts totally different. The forces we see around us, are only aspects of one over acting force - special limit cases; and depending on starting conditions, those forces would have been totally different or not present at all.
But that's neither here nor there.
So, based on observations, it appears that the Mirror Universe and the Federation Universe -- which are both EQUALLY REAL, in case there was any question -- each follow the basic laws of science on their own (if you ignore the existence of the other Universe).
That is, it makes perfect sense for Captain Kirk's parents to have a son, and for that son to become a starship captain, and to have Spock and Uhura and Sulu and McCoy on his crew, regardless of what Universe they are in. None of those events, happening once, violates the laws of physics or biology or thermodynamics or Chaos Theory.
But then you have to apply Chaos Theory over time. I believe the theory states that small changes, over time, inevitably lead to bigger changes.
That only works if you have nice starting point, and you know all variables, and then you can project that into the future. However, this is totally meaningless when it comes to past events. The past events are the past events, whatever those events are, or what physical laws govern them, matter not.
I saw a Ray Bradbury movie on TV last week, where some time travelers went back to see dinosaurs, someone accidentally stepped on a single butterfly in 65 million B.C., and that screwed up all of mammalian evolution by the time they got home, so that all of human society started to disappear. A small change in history, over time, led to a major change at the genetic level in the future.
This Butterfly Effect is a natural extrapolation of Chaos Theory, as it applies to time travel stories. For example, in "Star Trek: First Contact," the Borg went back in time to 2063 and killed Zefram Cochrane, and 300 years later we saw an assimilated Earth and there were no humans alive anywhere in the Galaxy. That means that after Cochrane's death, there would never be a Kirk or Uhura or Spock or Sulu or Picard or Data or anyone else whom we've seen in the original timeline.
In "Parallels," we saw many different timelines that all branched off at different points in history, and as time went on, each timeline became more and more different from any other.
In Isaac Asimov's "Foundation" series, it was stated that future history could be calculated for thousands of years in general -- i.e., which Galactic 8Empires would rise and fall, which political or social groups would gain power, etc. -- but small things like an individual person's name or genetic code could not be predicted even a few decades out, because those small-scale traits are "random" and unpredictable.
Rather off topic, but anyway: that is of course mutually contradictory. If you can't predict one individual, then you can't predict large scale social groups, because that one individual you can't prodect can completely alter the course of the large scale social groups.
Way to long to quote, but here's the answer to the rest:
No, it's not. You see, you make the mistake of demanding that the two universe were once one, and they must therefor split apart is logically flawed. You're making a premise without backing it up. What you're doing is a probability calculation upon a tiny sub-set of variables, based upon an uprovable premise; that they are once one, and thus they must split apart. Even if you say you don't; you do it implicitly. You apply your knowledge of physics and the implicit expressed changes of certain events happening because of those physics after the fact that these events happened and then say they can't.
Just us being here, having had the string of events that lead to us, have an equally low chance of happeing; aka 1 in a billion trillion zillion; or however small it exactly is. You'd say WE can't have happened then, without some supernatural interference; but this is ridiculous. EVERY sequence of events had an equal chance of ocurring, and ONE had to have occurred, this just happens to be the one that occurred. Now you can do that a second time, with a whole new set of events, these events could be different, they could be the same, they could be partially the same, but these has an equally low chance of having happened. Yet they still would have happened, without any supernatural inteference at all.
Or in other words: take a dice throw it a billion trillion zillion times, and write down the sequence. You'll find that the chance of thise one sequence occuring is only 1 in a billion trillion zillion. Does that mean a supernatural force willed that sequence into existence? Of course not. Now do it again, throw the dice anoher billion trillion zillion times. What are the chances of you have thrown exactly those two sequence of events? 1 in a billion trillion zillion times a billion trillion zillion. Does that mean a supernatural force made you throw those two sequences? Of course not, you went to throw two sequences, so you must have had two sequences at the end, you just happened to have thrown those two sequences. And the same laws of physics govern both sequences; now you check the sequence, and you find that a few patterns, or sub-sequences repeated themselves in both sequences. Does that than mean that a supernatural force made you throw the same sequences? Of course not; you just happened to have thrown those sequences.
This is the same with the universe and the mirror universe: you got two sequences of events. Certain parts are highly similar (but not actually the same), does that mean a supernatural force forced or chose those two universes? Of course not; they just are, it just happens to have been these two universes and they contained a few similar patterns along the line. You can't decide after a sequence of events have occurred, then arbitrarily go back and chose a starting point force feed some physics and chances in there, and say it can't have happened, so there's some unseen force. They just happened to have happened, and ran for a while parallel.
They just ARE.