Thanks. I don't know about talented; it's just that I have been around for a fairly long time.

The cohomology of $j! \mathbb{Z}$ is not the cohomology of the open subset. If $X$ is compact, it is the cohomology with compact support; in this way you can get Mayer-Vietoris for Borel-Moore homology.

This works for finite closed covers, not for open covers.

Metha and Srinivas wrote a paper on this topic, "Varieties in positive characteristic with trivial tangent bundle", <archive.numdam.org/ARCHIVE/CM/CM_1987__64_2/CM_1987__64_2_1‌​91_0/…>.

Yes, for $n=1$ the two spaces coincide. Let me add that the reason why $\mathrm H^{2n^2-1}(GL_n(\mathbb C)) = 0$ is that $GL_n(\mathbb C)$ is homotopy equivalent to the unitary group $U_n$, which has dimension $n^2$. Or, $\mathrm H^{i}(GL_n(\mathbb C)) = 0$ for $i > n^2$ because $GL_n(\mathbb C)$ is an affine variety of dimension $n^2$.

Another equivalent condition, which is sometimes useful, is that there are no non-zero quadratic differential that are invariant for the action of the automorphism group.

By a transitive action I mean that the morphism $\underline{\rm Aut}_S X \to X$ coming from the section is scheme-theoretically surjective. Since the fibers are connected, this implies that the connected component of the identity must dominate each fiber, and this is enough to conclude.