Thanks! This looks very much like what I was looking for. I also didn't know about Deligne's appendix to Residues and Duality that the work references.

The reference is the same two chapters of EGA (see I.10.11 in particular). I think the definitions there are equivalent to saying that a sheaf is a sheaf of topological modules which is $I$-adically complete and has locally a fundamental collection of neighborhoods of zero consisting of submodules. A sheaf is coherent (again in my re-interpretation) if it is locally a quotient of a finitely-generated free module by an open submodule.

Thanks Peter! This looks very much like what I'm looking for (and yes, the question comes from thinking about equivariant homotopy theory)

You don't need to be careful about punctures unless you care about complex-analytical type data at the punctures. But (perhaps this is nitpicking), you do need to be careful about "instability". When $C$ is anything but a sphere with no punctures, the character variety parametrizes based maps $C\to BG$, i.e. principal $G$-bundles with chosen trivialization at the base point. (For $GL_n$, this is equivalent to vector bundles: in general, there is a standard dictionary, see en.wikipedia.org/wiki/Frame_bundle). When $C$ has $\pi_2 \neq 0$ with $G$ not discrete, the two may not be the same

You don't need quasi-projectiveness. Being of finite type is enough: given any (irreducible) finite-type $X$ with $G$ action, let $Y\subset X$ be a closed divisor such that the complement $X\setminus Y$ is affine. Then the complement $X\setminus G\cdot Y$ is an affine scheme, and Sean's answer applies.

Let us continue this discussion in chat.

Right. (I removed my earlier comment about algebraicity and edited the answer instead). Re your second question, if I remember correctly, the category of extensions of an algebraically closed field $k$ of transcendence degree 1 is equivalent to the category of closed curves with finite morphisms, and you can use that there are no nonconstant morphisms from $\mathbb{P}^1$ to a curve of higher genus.