Just some 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.  

3. As you indicate, Infinity is indispensable, since $V_\omega$ is a model of all the other axioms, and even if the background theory is full ZFC (i.e., if we pretend to live in a universe satisfying ZFC), from $V_\omega$ we cannot build a model of ZFC.

4. I have to confess that I am not sure what you mean by the Abstraction Scheme.  Is this
what I would call Separation?  Anyhow, given an uncountable regular cardinal $\kappa$ (such as $\aleph_1$),  
$H_\kappa$, the collection of all sets whose transitive closure is of size $<\kappa$, 
is a model of ZFC without the Power Set Axiom  (I am again assuming ZFC as my background theory).  It follows that there is no procedure to build a model of ZFC from a model of ZFC without Power Set.  So it seems that any minimal subsystem of ZFC would have to include the Power Set Axiom.  

Partial Conclusion so far:  You need infinitely many axioms by Joel's answer, you don't need AC or Regularity, but you do need Infinity and Power Set.