Can PA+Con(ZFC) prove every theorem of ZFC?
No, since, as others have pointed out, ZFC is about sets, while PA is about integers. The language of ZFC can be translated into arithmetic, in the manner you suggest, but then PA is proving things about provability in ZFC, not proving the truth of the theorems themselves. For instance, if $ZFC\vdash \phi$ then already PA by itself (without Con(ZFC)), and even weaker theories, can prove the formula representing "$ZFC\vdash\phi$".
However it is true that if ZFC prove a $\Pi^0_1$ statement about natural numbers then PA+Con(ZFC) proves the same statement.