The only thing that confuses me (and I know it's something you just have to accept) is that if everyone has a doppelganger on each planet, how does anyone throughout all of history have a different fate than their doppelganger? Everyone has to have at the very least mated with the same people they mated with on the other planet, at the same time, under similar circumstances, and they had to live at least long enough to have the same number of offspring.
Yet the differences between the two worlds are so vast, not only technologically and historically, but socially as well, that it's pretty much an impossibility for there to be even a single doppelganger.
Well the same applies even moreso to the Mirror Universe on Trek. As you said at the beginning, it something you just have to accept.