What is the axiomatic (set theory) context of the P vs NP and NP=EXPTIME conjectures?
It's not specified. When there is a serious enough candidate paper purporting to resolve P ≟ NP, a Special Advisory Committee will be formed to decide whether (and to whom) to award the prize. I presume that the Special Advisory Committee will decide whether your system of axioms is acceptable. If you assume Z-F with choice, I guarantee you they will take it. If you assume P ≠ NP as an axiom, I guarantee you they won't.