Oh, ok. Hypercovers won't work? You alway have cohomological descent for those.

In this case, you should have the usual spectral sequences linking Čech cohomology and ordinary (sheaf) cohomology. This should imply that the $H^1$'s are the same; but it would surprise me if you can prove degeneracy without any local condition. On the other hand, it is known that the Čech to sheaf cohomology spectral sequence for open covers degenerates for paracompact spaces (I forget what the result is called, you can find it in Godement's book on sheaves). This is also very surprising to me. Maybe analyzing the proof of this result one could get some ideas for your case too.

Almost any book that treat UDF's, for example, Zariski and Samuel, or Matsumura's "Commutative Ring Theory" (Theorem 20.8). Anyway, $k[[x,y]]$ is a regular local ring, and those are well known to be UFD's (the Auslander-Buchsbaum theorem, 20.3 in Matsumura).

(continued from above) About my first sentence, sorry, I thought I meant something, but I am not sure anymore. In any case, I have seen your notion of $\infty$-finite type somewhere (I can't recall where, maybe Borel's book on D-modules), and it seems to me that it should always give an abelian category. Also, sorry again, "the sheaf of D-modules" in my second sentence should be "the sheaf D of differential operators".

Dear James, sorry for being unclear. I was trying to make a couple of points. The first is that the useful notion, in the contexts in which I work, is that of finitely presented sheaf, not that of coherent sheaf. On the other hand, essentially all you use is that locally they come from noetherian schemes, and thus all the real work is done with noetherian schemes. The second is that the coherence condition is important in other contexts, for example when working with D-modules. In my second sentence I was just trying to give an example of when it is important (continued below).

That's exactly right.

The component of the Hilbert scheme is determined by degree and genus; infinitely many genera means infinitely many components.

I strongly believe that there should be an upper bound for Cohen-Macaulay (notice that reduced curves are automatically Cohen-Macaulay). Of course for proving boundedness this is not good enough, but if the OP cares I might try to write up a proof.

Does this work for non-reduced curves?