Same relative logic, just more of it/more steps or trials.
Yes, and that's exactly what they do.
No, none of the problems you gave to the LLM while toying around with them are in any way novel.
I think you are confused about LLMs - they take in context, and that context makes them generate new things, for existing things we have cp. By your logic pianos can't be creative instruments because they just produce the same 88 notes.