Is every "nice" abelian category with enough projectives an additive presheaf category? A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of increasing restrictiveness)


*

*ABn for some $n$.

*Grothendieck

*locally finitely presentable

*the category $\mathsf{QCoh}(X)$ of quasicoherent sheaves on a scheme $X$ (possibly with further adjectives)

*etc.
In particular: if $\mathsf{QCoh}(X)$ has enough projectives, then is $X$ a disjoint union of affine varieties?
Clarification: Just to be clear, I'm well aware of the Freyd-Mitchell embedding theorem. This is not a question about how close abelian categories are to module categories -- it's a question about how restrictive it is for an abelian category to have enough projectives. The local presentability hypothesis rules out the duals of categories of sheaves for instance.
Motivation / Evidence:
I'm thinking of things like this result: the category of sheaves on a locally connected topological space has enough projectives iff that space is an Alexandroff space -- a very restrictive condition.  I suspect that the category of sheaves on an Alexandroff space is a  module category  additive presheaf category.  Although on reflection, the category of sheaves on an Alexandroff space need not be an additive presheaf category -- in particular, it need not be locally finitely presentable. So perhaps one should assume that $\mathcal C$ is locally finitely presentable for the purposes of this question.
For another example in this direction, consider the fact that the category of quasicoherent sheaves on a smooth projective variety of dimension >0 over a field never has enough projectives. (See my CW answer below for a more general result).
Alternative formulation:
If we assume that $\mathcal C$, in addition to being a "nice" abelian category with enough projectives, has a compact generator, then is $\mathcal C$ a module category? This would follow from the formulation above, since an additive presheaf category with a compact generator is a module category; but it could conceivably be easier to show. It would also settle a form of the question about categories $\mathsf{QCoh}(X)$.
 A: The category $[C^{op}, \text{Ab}]$ of $\text{Ab}$-valued presheaves on any (small, for simplicity) $\text{Ab}$-enriched category is about as nice as it gets - locally finitely presentable, Grothendieck, etc. - and all coproducts of representables are projective (these are the "free" objects), but it won't be a module category in general if $C$ has infinitely many isomorphism classes of objects. 
The simplest example is to take $C = \mathbb{N}$: $\text{Ab}^{\mathbb{N}}$ has the property that any generator must be supported at every element of $\mathbb{N}$, and no such presheaf can be compact. In fact the compact projectives are precisely the finitely supported sequences of finitely generated free abelian groups. Note that this is also the category of sheaves on an Alexandrov space, namely $\mathbb{N}$ with the discrete topology. 
A: Let $\mathcal C_0$ be a small additive category with countable coproducts, and let $\mathcal C$ be the category of additive functors $\mathcal C^{op}_0 \to \mathsf{Ab}$ preserving countable products. Then by Lemma 1.3 and Prop 2.5 here, $\mathcal C$ is a cocomplete abelian category which is easily seen to be locally $\aleph_1$-presentable. Moreover, cokernels are computed "levelwise", so that the representables are a generating set of projective objects; in particular, $\mathcal C$ has enough projectives. But $\mathcal C$ is pretty far from being an additive presheaf category.
Along with მამუკა ჯიბლაძე's example in the comments of sheaves on a complete boolean algebra, this makes it pretty clear that there are plenty of locally presentable abelian categories with enough projectives which are not additive presheaf categories. I'm still not sure about categories of quasicoherent sheaves, though.
