Yes, $2^{2g}$ of course.

Assuming X nonempty, this map is proper iff G is proper (I am assuming everything is over some base field k). Indeed, the map is a G-torsor, so its fibre above a geom pt of [X/G] is G.

The "good locus" is a finite etale double cover (outside characteristic 2).

Maybe that one is (better: has the structure of) a $K/K'$-torsor over the other? This makes sense regardless of whether the spaces are connected.

Freeness ensures that $X\times_{X/G} X \simeq G\times X$, so that descent data for $X\to X/G$ correspond to $G$-equivariant structures. Then flatness of $X\to X/G$ gives you effective descent. If the action is not free, you can take the quotient stack $[X/G]$ and then the equivalence in question is (basically) a tautology.

@PeterMueller That's problematic since finite index subgroups might not be open if the group is not topologically finitely generated.

The local maps $X\to U$ need not be unique (otherwise they would glue to a global map $X\to U$, and then $U\to\mathcal{M}$ would be an isomorphism).

I think it means that the moduli stack $\mathcal{M}$ (the "functor" associating to every $S$ the groupoid of suitable stable sheaves on $S\times X$) admits an etale surjection from a scheme $U$. That is, a natural transformation ${\rm Hom}(-, U)\to \mathcal{M}$ (corresponding by Yoneda to a suitable family of stable sheaves parametrized by $U$) such that every map $X\to \mathcal{M}$ etale locally on $X$ lifts to $U$. This means that every family parametrized by $X$ is etale locally on $X$ the pullback of the family on $U$ via some map. The map to the coarse moduli space $M$ need not be etale.

Is it possible that the stacks are the same but the maps to $BG$ ($G=\mu_2$ in your example) are different?

@LaurentMoret-Bailly good point! Yes, either that or Artin approximation for coherent sheaves.

Suggested edit for clarity: "Then the image of every fully faithful functor $\mathcal{T}'\to \mathcal{T}$ is admissible." Otherwise there is no relationship between $\mathcal{T}$ and $\mathcal{T}'$. Or/similarly "in the geometric case the fully faithful functor" replace "the" with "every."

@Aphelli yes. In my defence, I am not the only one using "finitely generated" to mean "topologically finitely generated" in the context of pro-finite groups.

@Sasha my notes say it's in Bloch-Kas-Lieberman "Zero cycles on surfaces with $p_g=0$" (Compositio 1976). But now looking at that paper I am not so sure anymore. They do construct a correspondence between an Enriques and the relative Jacobian of its elliptic fibration, and then show the latter is a rational surface (presumably it then has to be the blowup of $\mathbf{P}^2$ at the intersection of two cubics?). But the paper is about zero-cycles and the statement is preserved by birational equivalence, one would have to check what it says about Chow motives.