If positively graded means non-negatively graded then $n\geq 3$ can be replaced by $n\geq 1$.

Yes. That's right. I'll change it.

I think you need to be more precise about what kind of conditions you are looking for. Otherwise, all the entries below the diagonal are zero would seem to be a correct answer to your question.

Surely $[A,B]$ doesn't makes sense unless $p=m=n$?

That's right.

Kazhdan--Lusztig conjectures? en.wikipedia.org/wiki/Kazhdan%E2%80%93Lusztig_polynomial

Why should the age of a paper harm its quality as a reference?

ncatlab.org/nlab/show/Hochschild-Kostant-Rosenberg+theorem

Not really engaging with your question, but a family of examples over a field $k$ of characteristic $p$ (for $p$ odd) is the group algebra $kG$ for $G$ any group of order $2p^n$. The Sylow $p$-subgroup of $G$ must be normal as it has index $2$ and must act trivially on any simple module. Thus the simple modules factor through $kC_2$ which obviously has two (1-dimensional) simple modules.

Over $\mathbb{C}$?

Thanks. It wasn't so much that I would expect them to be there as that I thought it plausible that they might be.

Looking again I guess you're meaning Proposition 4.7.2(1) which does point in the right direction in that it explains the relationship between the locally free sheaf and the gluing data of the trivial geometric sub-bundles. I'd like something more explicit though.

Sadly not. They do talk about locally free sheaves and call them vector bundles but they don't discuss what I call geometric vector bundles in my question.

Could I ask whether this paper of Köpf answers my question mathoverflow.net/questions/121881/…? It doesn't seem to be so easy to obtain a copy.

No, I believe that there are no extra difficulties once you have proved that the sections of the trivial geometric vector bundle naturally form a free module over the global sections. A citeable reference to a general statement for ringed spaces (when the base space is equipped with a Grothendieck rather than usual topology) would also be appreciated if one does not exist for this specific case --- the fewer things I would have to explicitly check the better.