Reply to sibling - lean4 doesn't rest on ZF or ZFC. https://lean-lang.org/theorem_proving_in_lean4/Axioms-and-Co... However I believe an equivalence of power has been shown between the two.
Roughly, yes. See B. Werner (1997) “Sets in types, types in sets”.
Roughly, yes. See B. Werner (1997) “Sets in types, types in sets”.