Well, wormholes are the most popularized, but there are other solutions that have closed time-like curves that don't involve wormholes. In these, you'd just be flying along, and when you finished flying, you'd find you landed before you took off.
And, conversely, not all use of wormholes end up with closed time-like curves.
I didn't mean to imply that wormholes MUST have 'time-like' world lines. I'm not sure about 'other solutions', there are a LOT of different formulations of GR, including rotating space time that could produce various results. Wormholes are at least the one that I know of that is localized. In any case they are probably impossible.
"Travel to a place that is a lot like what my world 3 weeks ago," doesn't really violate anything. In and of itself, it is just going to another planet. However, if you go there *faster than light could get there*, then we have the same problems as any other travel faster than light. The nature of the destination isn't the issue, it's how fast you get there that matters.
I'm not so sure about that. Remember, we're talking about a part of the Universe which has never been in communication with the part we inhabit since the start of inflation, no photons have ever been exchanged, and none ever could be. So, if you took some action somewhere in that space time which would violate causality, etc in OUR space time's past, no observer would ever be able to tell, because they inherently cannot compare the events in the two places, there isn't any light cone anywhere that overlaps both.
I think you would have to have some inertial frame of reference in which an actual observer could exist who would observe some violation of causality or some conservation law in order for a 'problem' to exist. Because the two areas of space time don't interact with each other or any other reference frame that interacts with either one of them, a sort of loophole is created. Such areas of the Universe, because they cannot exchange information with us, don't really exist in some formal sense as far as we're concerned. At least one COULD look at it that way. Obviously we don't know what is or isn't possible, and we haven't even a clue how one would open up a 'pathway' of some sort to such a remote location. Its just as likely such considerations are moot because it simply isn't possible.
Note that, as soon as you go to a place, that place is causally connected to Earth. You came from Earth, and you can now cause events there.
As to how they would become connected going forward, sure they would, but so what? There's no 'paradox' there. If you kill 'alternate grandfather' it will surely mean that 'alternate you' won't come into existence (or whatever) but 'prime you' still has a grandfather, prime grandfather. I would at least vote for their being no violations of conservation laws either, because, as you say, the two areas of space time ARE now connected. Still, for your own purposes, if you wished to 'go back in time and kill your grandfather' you can now fully enjoy the benefits of having done so, as long as you aren't concerned about y