This is actually the correct application of Many Worlds...
Not really. MWI doesn't require that every imaginable reality actually happens; it just says that many of them happen. And of course the laws of physics and probability still apply; only things that are physically possible can happen, or things that can validly result from a given set of circumstances. There are lots of possible timelines that can be imagined but that couldn't actually happen without violating those parameters.
Not to mention that it's a conceit of fiction that the differences between timelines are based on macroscopic, human-scale events like personal choices. In reality, the divergence is brought about by quantum-level effects, and since macroscopic events follow classical physics, quantum divergences between timelines would probably average out to irrelevancy once you got the macroscopic scale, leaving you with an ever-increasing number of parallel quantum timelines that were essentially identical on a scale a human observer could perceive or care about. There would only be certain contexts wherein a quantum-level change could have any effect on macroscopic events -- something like the setup of the Schroedinger's Cat thought experiment where a mechanism is rigged to release upon detecting a quantum-scale event like the decay of a radioactive particle. Maybe in some instances a radiation-induced mutation might happen in one timeline and not another, triggering a small but increasing divergence between species in the two. And once civilization got to the point of inventing technologies based on quantum effects, like transistors and diodes, then quantum differences might begin to have more of a noticeable effect on a human scale. The invention of quantum computers would make it happen even more often. So you'd have a bunch of timelines that were macroscopically identical for most of prehistory, aside from the occasional sheaf that branched off with a different evolution, but once a civilization arose and reached the computer age, the timelines might start to diverge more.
It may be unsatisfying but it sure as hell makes more sense than Voyager's time travel mechanics. Universes don't pop into existence when they branch. Pre-existing but overlapping timelines simply diverge.
Err, no, I don't think that's how it works. At least, not in the sense you mean. In any case, the distinction is irrelevant on the scale of personal experience and events, because we're still talking about a history that was identical until Nero came back, and that's very different from the Mirror Universe, which was different from a far earlier point.
They seem to be forgetting that, as established by "Broken Bow," it was Zefram Cochrane who originally gave the "explore strange new worlds" speech. Although I guess it's possible that McCoy could be unfamiliar enough with the history of spaceflight to be unaware of that.