The Many Worlds Interpretation Does Not Apply to Macroscopic Time Travel Effects
The Many Worlds Interpretation is very specific in its meaning and application. It only applies to probabilistic quantum wavefunctions, such as radioactive decay and the trajectory of a photon. It does not apply to macroscopic events, except in when macroscopic events are incluenced by probabilistic quantum events, which is rarely.
In the case of time travel, the probabilistic wavefunction is the "light cone" of the time traveler, the exact position and trajectory at which he enters the past. The Many World Interpretation that for every position and trajectory in the light-cone there is a universe in which the time traveler came into the past at that point with that trajectory. However, each of these worlds is entirely self-consistent, with the time-travel effects both pre-written and unalterable.
The self-consistency principle further limits the potential light cone of a time traveler because it cannot create inconsistent macroscopic effects, which means that Reese must be in a position to rescue and impregnate Sarah Connor in every world, the light-cone simply does not extend to positions where he cannot do so.
Interesting. I'm not sure I understand all of that.

Let me see if I've got it kinda right.
So you're saying that quantum probability only applies to very small sub-atomic/photonic events. Meanwhile, larger stuff like me eating a cheeseburger tonight, a meteor impacting the Earth, or Kyle Reese impregnating Sarah Connor can only occur one way and occurs the same way even in all of the small, variated universes that might be created by the different outcomes occurring on the quantum level. Am I on the right track here?
Not exactly. The Many Worlds interpretation denies the reality of waveform collapse and replaces it with the mechanism of decoherence. Take the Schrodinger's cat thought experiment, in which a poison will kill a cat based on the state of a quantum probabilistic event. Without the MWI, the cat is said to be alive and dead simultaneously until waveform collapse. In MWI, it is said to remain simultaneously alive and dead after decoherence, we just only interact with one state or the other.
In regards to time travel and consistency, the best example is the billiard ball through a wormhole example. Lets say we have a lone billiard ball sitting on a pool table and a wormhole opens up for no apparent reason. Out of this wormhole comes the billiard ball's future self, which strikes it causing it to fall into the wormhole and strike itself in the past.
When the wormhole opens up initially, there are an infinite number of trajectories that the future billiard ball could have when it comes out, and an infinite number of trajectories would allow it to hit it's past self into the wormhole in such a way as to create a stable loop. The Many Worlds Interpretation says that each of those probable trajectories for the future billiard ball continue to exist side-by-side after decoherence.
The important thing is that decoherence occurs when the ball comes out of the wormhole in the past, not when it enters the wormhole in the future.
This means that there are an infinite number of ways that the time travel incident could play out. All of these ways exist simultaneously, and all are self-consistent and
none interact with any of the others. You can't create a new worldline by going back in time because the decoherence caused by your appearing in the past has already occurred. The new worldlines already exist, you're just unaware of them.
In the case of T1, there are an infinite number of self-consistent ways that the time travel could have resolved itself. Most of these would likely involve Reese impregnating Sarah, probably all of them, though the details could be wildly different. There could, for example, be a timeline in which Kyle survives and raised John with Sarah. There could even be one where Kyle isn't John's father. But, and this is a big but, we didn't see any of those other timelines. There might even be a timeline in which no one time travels at all, but that can't be said to the the original timeline, because no one from it's future ever steps into a time machine.
When most people try to use Many Worlds to reconcile time travel, they make the mistake of thinking in hypertime. They reason that traveler from A goes back and caused B to branch off, and continues on in B. There is no hypertime.
What happens is that a time bubble appears in 1984, at some spot. The exact spot where the bubble will appear is determined by quantum probability, though self-consistency demands that it appear close enough to Sarah Connor for the time traveler to rescue her from the Terminator (whose on time bubble has already appeared at a spot also determined by quantum probability). For every possible spot that the time bubble could appear, a timeline comes into being and a traveler from that timeline's future comes through. Most of those are Kyle Reese, probably all, though some may not be, and all are playing their part in a self-consistent predestination paradox. Some of futures are similar to the one we know, others may be radically different. This doesn't matter, because all these timelines are segregated from each other by quantum decoherence, and time travelers from one are not crossing over into another. Each has it's own self-consistent timeloop, determined by where the time bubble appeared when it came into existence in 1984, with was determined by quantum probability.
The key is that although the timeliness do not have to be the same, they don't have to be even remotely similar, they must all be self-consistent.
What about totally independent parallel universes slowly branching off through small different outcomes? In essence, could the Mirror Universe from Star Trek exist parallel to our own, even if it is inaccessible to us? Or is that just wild fantasy?
That's pretty much what the MWI is about, though it isn't exactly parallel universes. The MWI describes a quantum superposition of states that always exists, it's just that deconherence only allows us to be aware of one. It's all one universe, with all the same matter and energy, just in a different state.
Exactly how much these states can diverge is a rather important question, and one that is up in the air. Quantum probabilities are additive, which suggests infinite potential divergence on a quantum level. However, most probabilistic quantum events have little effect on the macroscopic world, and they tend to occur in such great numbers that they all even out.
It is unknown exactly what effects quantum events have on human thought processes. It could be significant, or it could be none at all. If probabilistic quantum events did effect human thoughts and decision making in a significant manner, then thus would eventually lead to extreme divergence, possibily of the mirror universe type.