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.
Yes, in the following sense: If $T$ is a first-order theory which happens to have a classifying topos, then it is Morita-equivalent (has naturally equivalent categories of models in any Grothendieck topos) to a geometric theory. This is because any Grothendieck topos is the classifying topos of some geometric theory. (Here I'm following the usual terminology in the topos theory community to allow arbitrary disjunctions in geometric formulas, else the latter statement is wrong.)
(Since the term "classifying topos" is not usually defined for first-order theories, let me be more specific: Suppose that, unusually for first-order theories, pullbacks of models of $T$ along geometric morphisms are again models of $T$. Then we can setup a (pseudo-)functor associating to any Grothendieck topos $\mathcal{E}$ the category of $T$-models in $\mathcal{E}$. We say that $T$ has a classifying topos if and only if this functor is representable.)
Not as usually stated, for exactly the reason you mention; same with IZF (intuitionistic Zermelo–Fraenkel).
But I believe that Malice's observation in the comments is right: The Morleyization of ZFC is a geometric theory, and ZFC is Morita-equivalent to its Morleyization.
However, note that in many toposes, there will not even be a single model of ZFC (or its Morleyization): From such a model we can construct a model of PA. But many toposes don't contain a model of PA, for instance the effective topos doesn't (assuming any metatheory) and sheaf toposes over overt spaces don't (assuming a metatheory which doesn't provide for a model of PA). Also, assuming ZFC as the metatheory, not even the topos of sets can be verified to contain a model of ZFC.
No, that's false, even in the case that $\mathcal{E}$ is the topos of sets. It would be true if instead of $\mathrm{ZFC}$ we put $\mathbb{O}$, the theory with exactly one sort and no atomic propositions, relations or axioms whatsoever.
A model of ZFC (in the topos of sets or in an arbitrary topos) is not (what the topos believes to be) a single set, but (what the topos believes to be) a full set-theoretic universe.