There are two parts to my question:
Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?
My reasoning is thus:
It is well-known that there is no non-trivial category that is both a topos and an abelian category -- since the topos is cartesian closed, it cannot have a 0 object without becoming trivial (see here: Can a topos ever be an abelian category?).
However, by the affine Serre theorem, it is well-known that the category of $R$-modules is equivalent to $QCoh(Spec(R))$, or the category of quasi-coherent sheaves on the space $Spec(R)$. Since the category of $R$-modules is abelian, it follows that the category $QCoh(Spec(R))$ is not a topos despite the fact that it is some kind of category of sheaves (satisfying some extra property). Am I correct?
Question #2) If this is true, can someone give me a conceptual understanding of why one would wish to present the category of $R$-modules as the category of quasi-coherent sheaves, even if it's not a topos?
I get the rough idea that we're constructing the category of $R$-modules from the categories of modules over local rings $R_{\mathfrak{p}}$ -- but I'm not sure I understand this very clearly. Can someone explain this a bit more clearly? Where does quasi-coherence figure in this picture?
Further, suppose we have a category of sheaves over the site $Spec(R)$ endowed with the Zariski topology, which is a topos (right?). Is the internal category of $R$-modules within this topos equivalent to the category of quasi-coherent sheaves? Or is that not the right way to think about the category of quasi-coherent sheaves? (Perhaps the answer is sitting in Ingo's PhD thesis somewhere, but I haven't found it...)