how many empty sets are there?

Your conclusion is not entirely correct. While it's perfectly fine to have a context where a category has many initial objects which are isomorphic (but not equal), there is a delicate point to the inaccurate statement:

the class of say, all elliptic curves over $\mathbb{C}$ is a proper class, but the class of $\mathbb{C}$-isomorphism classes of elliptic curves is a set. That would seem to imply that every isomorphism class is a proper class

The fact that there is a proper class of different structure is quite trivial to show, because we can always change one element with another and get a proper class of different, but isomorphic, structures that way.

To say that there is only set-many isomorphism classes is also inaccurate, because the isomorphism classes are proper classes, there is no collection of the form $\{A\mid A\text{ is an isomorphism class of ...}\}$. Only sets can be members of other sets. But it does mean something else, it means that there is a set of pairwise non-isomorphic structures that any other structure is isomorphic to one of them. That is, there is a class of representatives which is a set.

Lastly, it does not mean that every equivalence class is a proper class, just that at least one of the equivalence classes are.

The simplest example of this is equinumerousity (in $\sf ZFC$). There is a proper class of sets of every cardinality, except for the class of sets of cardinality zero. There is only one of those.

If your foundational theory is $\sf ZFC$ or some related theory then there exists only one empty set. If your foundational theory is some category based theory which allows many empty sets, but requires them to be isomorphic, then this is a different case altogether.

By extensionality axiom you get that there exists exactly one empty set, and that's all folks!