Given a language $L$, the set of geometric formulae in $L$ is the smallest set of formulae containing atomic formulae and closed by finite conjunction, arbitrary disjunction, and existential quantification.
Recall that a theory is geometric if it can be axiomatized by sentences of the form $\forall \mathbf x (\phi(\mathbf x)\rightarrow \psi(\mathbf x)$ where $\phi$ and $\psi$ are geometric formulae and $\mathbf x=(x_1,\dots,x_n))$ for some $n$.
1) If $T$ is a geometric theory then it has a classifying topos. Does the converse hold? I.e., does the existence of the classifying topos imply that the theory is axiomatizable as a geometric theory?
2) Is ZFC a geometric theory? I suspect it is not, since it would seem to me that implication and double implication cannot be axiomatized with geometric formulae, and therefore we have problems since $$\forall x \forall y (\forall z (z\in x\leftrightarrow z\in y)\rightarrow (x=y))$$ (extensionality) doesn't seem to be re-expressible by means of geometric formulae (in part., I have trouble in converting $\forall z( z\in x\leftrightarrow z\in y)$ into a geometric formula.) Or am I missing something? Mind that I am just at the beginning of my studies in this topic.
3) Is it true that $Mod(\mathcal E, ZFC)=\mathcal E$? The only reason I have to believe this is that it should be true (correct?) if $\mathcal E=Sets$, so I wouldn't be surprised if the answer is no in general.
Also, if point (1) is true and point (2) is false, also point (3) should be false, since if it were true we would have $$Geo(\mathcal E, S(U))=\mathcal E=Mod(\mathcal E, ZFC)$$ where $Geo$ means topos morphisms and $S(U)$ is the object classifier; and hence $ZFC$ would have a classifying topos, contradiction.