Technically, if you did manage to change time, whether in a single timeline universe or a multi-verse, you should blink out of existence upon success, because the you there at that time would either not exist at all (single universe that had been changed), or would be existing in a universe it was impossible for you to be in (multi-verse with a new timeline just created which does not include you, or includes an alternate version of you native to that timeline). Time travel stories usually bend the rules a great deal in order to imagine that the protagonist can still exist, or retain memories of things before the change, etc.
I don't think your technical explanation really makes all that much sense. If you are able to change time, it doesn't necessarily imply that you will all of a sudden cease to exist. The whole idea behind the multiple worlds theory is to get around the idea of such a paradox.
Exactly. In my humble opinion, the multiple worlds theory is lazy thinking created by people who cannot deal with the paradoxes of the Copenhagen interpretation of the results of quantum physics experiments.
If you go back in time and kill your grandpa, you (the killer) still exist in this timeline, but another version of yourself is never born.
I don't think that's correct.
In the traditional Grandfather Paradox, a single time universe is assumed. You are born. You go back in time. You kill your grandfather thus making it impossible for you to be born - you blink out of existence because you were never born. The question here is, since you blink out of existence, is it impossible to kill your grandfather, or can you kill him and only then blink out of existence?
Even in a fictional context the multiple universes theory is problematic, and here's why. You are born. You go back in time. Now, does this action create a new universe? One would assume yes since time travel should probably be "unexpected" and not a part of the natural timeline - unless that time travel was "meant" to happen thus maintaining a single universe through this point. But what determines what events are "natural" to a timeline? That's basically an unanswerable question.
Say the universe does not diverge at this point, but only at the point at which you actually change what had previously occurred - killing your grandfather. The minute your grandfather's heart stops beating, the universe diverges. Does the you who just killed his grandfather remain in the first universe, or does it continue on into the second? Do you suddenly split as well - in the original universe blinking out of existence and in the second going on? If you go on in the second universe - where did you come from within that universe? You weren't born there, so how can you exist there? Stories tend to assume it's easy to skip from one universe to the next, much like they assume that we could ever communicate with aliens - but its purely a story conceit. Every atom in your body would be foreign to the newly created universe in which you have no history. Where did the atoms in it even come from? How could they possibly exist within that universe?
Star Trek has, as usual, not been at all consistent on whether or not there's a single timeline universe, or a multi-verse affected by time travel. Several times the timeline has been put "right", implying a single universe. Several times we've seen alternate universes affected by time divergence. We've even seen a time travel branch of Starfleet bopping around. Do we want to presume they're splitting off new universes every time they act - even though there is never the slightest bit of discussion of this among the characters? Even though every single bit of dialogue, except perhaps in the episode "Parallels" has the characters assuming a single universe?
You're asking a question which really can't be answered because Star Trek is inconsistent on this score, and the actual logical consequences of either a single universe or multi-verse are never dealt with in fiction.