You mean that for any INFINITE set A, we have A equinumerous with AXA.

Personally, I don't see how to make your question precise. What is meant by "what determines whether"? I mean, if for a certain A, whatever axioms prove that A is equinumerous with AXA, then that determines it, but, of course, that doesn't preclude that one could provide different axioms to make the proof.

As to needing the full axiom, in any given case, we could use a weaker form of the axiom, even if just by stating the axiom except only with regard to the particular set, such as R or whatever set. So, at least to me, it's not clear what you mean by a "cut off point".