An abelian category is somewhat like an elementary topos, but there's a better analogy: the Barr exact categories. These are categories with finite limits and quotients of equivalence relations which interact nicely with pullbacks. Every topos is Barr exact, but not conversely-for instance, categories of algebraic structures are Barr exact but very rarely toposes, since they're rarely Cartesian closed. And indeed, Barr proved in his original paper on the subject that a category is abelian if and only if it's additive and (Barr) exact.
Let me next answer your question about embedding small topoi into sets. This is not again not quite the right analogy to make: we should talk instead about small Barr exact categories. And it's unreasonably to expect to embed them nicely in the category of sets, since for instance the terminal object would then be a generator. The closer analogy with Mitchell's theorem would be to ask that a small Barr exact category $\mathcal E$ embed exactly in a category of right $M$-sets for a monoid $M$, being the non-additive analogue of a category of $R$-modules. Unfortunately, this is still not possible: for instance, the terminal object in the category of $M$-sets has no nontrivial subobjects, while in $\mathcal E$ it might.
However, this is essentially the only constraint. In the same paper referenced above Barr exactly embeds every small exact category in the category of presheaves on a category built out of its poset of subterminal objects. The last caveat is that, outside of the abelian world, "exact" must have a new meaning. Specifically, it means "preserving finite limits and coequalizers of equivalence relations." It is an interesting exercise to see that this reduces to the usual definition of exactness for functors between abelian categories.
Regarding your question on a characterization of exact reflective localizations of categories of sheaves of modules, there is a well-known answer to a slight generalization. Let us allow $X$ to be preadditive and consider additive functors. Then such categories $C$ are exactly the Grothendieck abelian categories, that is, the cocomplete abelian categories admitting a generator in which filtered colimits are exact. One direction of this is the famous:
Theorem (Gabriel-Popescu) Any Grothendieck abelian category is an exact localization of a category $[A^{\mathrm{op}},\mathrm{Ab}]$ of additive presheaves of abelian groups on a small preadditive category $A$.
There is no need to consider a topology on $A$ here, as an exact localization of a category of sheaves is certainly an exact localization of a category of presheaves.
For the converse, it is a basic fact that reflective subcategories of cocomplete categories are cocomplete and that the reflection of a generator provides a generator. Finally, if $\mathcal{C'}\to \mathcal{C}$ is an exact localization and filtered colimits commute with finite limits in $\mathcal{C}$, then they do so in $\mathcal{C'}$. Indeed if $A$ is filtered then all the functors $$\mathcal{C'}^A\to \mathcal{C}^A\stackrel{\mathrm{colim}}{\to} \mathcal{C}\to \mathcal{C'}$$
preserve finite limits by assumption, while the composite of the above string is the $A$-colimit functor of $\mathcal{C'}$.
Best Answer
You make good points, and I am inclined to agree that the definition is strange. But it works, because if you have a classifying topos that works for all infinitary pretoposes then it also works for all cocomplete toposes in particular. If I recall correctly the concept of pretopos is not used in MacLane–Moerdijk, so maybe they say "cocomplete topos" instead as a compromise.
Incidentally, I believe the classifying toposes that arise in practice do work for all infinitary pretoposes. One way of justifying this is to observe that every small set of objects in an infinitary pretopos is contained in a (not necessarily full) subcategory that is a Grothendieck topos and such that the inclusion preserves finite limits, small colimits, and is conservative. Every "inverse image functor" from a Grothendieck topos will factor through such a subcategory (but the subcategory will depend on the functor), so the category of "inverse image functors" will be a union of categories of geometric morphisms between Grothendieck toposes. Similarly, every model of the theory you want to classify will also be contained in such a subcategory, so the category of models will be a union of categories of models in Grothendieck toposes.