Why exactly are complexity theorists interested in closed timelike curves?
Sorry for the very "big picture" answer from a non-quantum-theorist, but this contrast might help: you could describe algorithms as the study of how to solve computational problems, whereas complexity theory studies the resources (especially time) required in theory to solve them. How hard are these problems really at some fundamental level, and how are they classified by resource requirements? This question is considered interesting regardless of how many or what kinds of resources puny humans happen to have at their disposal.

(For instance, an $n^{100000}$ time algorithm for SAT would totally upend our view of complexity theory even if it never impacted any algorithms we could run in this universe. A $2^{0.00000001n}$ time algorithm for SAT would change our understanding of complexity much less, even though it would have huge practical implications.)

The question of how much time is needed in theory to solve a problem changes if you allow CTCs, therefore they are interesting to complexity theory by definition.

So I feel your first question is like asking why computability theorists would study Turing degrees higher than zero; it's what they do.