Just two minor points:  

1. The situation with the Axiom of Regularity (well-foundedness of the $\in$-relation) is similar to the situation with the Axiom of Choice.  In any model of ZF without regularity we can build the usual von Neumann hierarchy $V_\alpha$, $\alpha$ an ordinal, by iterating the power set operation, and the union of the $V_\alpha$'s is an inner model that satisfies regularity.

2. Depending on the precise formulation, the Replacement Scheme often implies the Separation
Scheme.