Note that all curves (over a fixed algebraically closed field say) are homemorophic in the Zariski topology. Hence you can take a quasi-coherent sheaf on one curve and then transfer it by a homeomorphism to another curve where it very rarely will come from a quasi-coherent sheaf. This makes it very unlikely that there is any kind of reasonable description of the essential image (other than the tautological one).

More concretely when, to make life simpler, we assume that $A=\matbb Z$. The torsor then is the disjoint union of $U_1\times 0$ and $U_2\times 1$. We have an action of $C_Y\times A$ on this. Over $U_1$ $U_1\times 0$ becomes $U_1\times 0$ and $U_2\times 1$ becomes $C_Y\times 1$ and on $U_2$ the opposite occurs.

The characteristic certainly plays a role in the answer. For all characteristics but two the invariant ring is Cohen-Macaulay but in characteristic two it isn't. It is of course true that you might find a (more or less) characteristic free presentation of the invariant ring where this fact is not apparent.

Note that the invariant ring is in general not Cohen-Macaulay so it is not free as a module over the symmetric polynomials. I think that excludes the possibility of a basis permuted by the group.

A reasonable amount of work seems to have been done (for any indecomposable representation not just a permutation representation). A starting point is (as well as the references to it in SciMath): MR0499459 (81b:14024) Almkvist, Gert; Fossum, Robert Decomposition of exterior and symmetric powers of indecomposable $Z/pZ$-modules in characteristic $p$ and relations to invariants. Séminaire d'Algèbre Paul Dubreil, 30ème année (Paris, 1976--1977), pp. 1--111, Lecture Notes in Math., 641, Springer, Berlin, 1978.

I don't know the actual etymology but I have thought it is as follows: In the (very) old days a singular (in the sense of special) modulus was the $j$-invariant of a lattice in $\mathbb C$ with complex multiplication. From the point of view of elliptic curves this corresponds to the $j$-invariants of complex elliptic curves with endomorphism ring larger than $\mathbb Z$. In positive characteristic there are some elliptic curves with larger endomorphism rings than complex curves can have, they are clearly even more special, hence supersingular.

Nice. It essentially covers what I did here.

Correcting the title is not just nitpicking. The paper is about a generalisation of the Hodge conjecture concerning a characterisation of the filtration on rational cohomology induced by the Hodge filtration. It deals with rational cohomology and so is not concerned with the failure of the (non-generalised) Hodge conjecture for integral cohomology. The latter result is due to Atiyah-Bott.