Algebraization of formal deformations is serious, even in the ordinary case (where it is affirmative, by an argument of Serre-Tate at the end of Messing's thesis). Beyond dimension 1 there are many non-algebraizable formal deformations over $W(k)$. The Norman-Oort paper assumes alg. closed $k$ (due Dieudonne modules), but this can be bypassed via a deformation ring argument (giving a lift over a $p$-adic order with residue field $k$, not that its normalization has that residue field!). Later work of Norman & Ogus gave obstructions, and showed a lift always exists over $W(k)[\sqrt{p}]$.

@Rupert: See 4.5 and 4.6 in Bruhat-Tits III (J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 34 (1987), 671--698); in 4.5 they work over a big class of valued fields and get some outer-type A examples too, and in 4.6 they specialize to finite residue field. They also note that Kneser had handled char. 0 local fields earlier. Hopefully there is a more accessible reference that someone else can provide.

For any two finite type separated DM stacks $X$ and $Y$ over a field $k$, a $k$-morphism $f:X \rightarrow Y$ is quasi-finite if and only if the induced maps between sets of (isom. classes of) $\overline{k}$-points has finite fibers. That is a basic exercise with the very definition of "finite type" for DM-stacks and ways of checking quasi-finiteness for finite-type schemes over a field. So that answers your question affirmatively since $\overline{k}$-points of the coarse space are the same as (isom. classes of) such points of the given separated DM stack.

For any (paracompact Hausdorff) complex-analytic space it coincides with twice the analytic dimension (for killing abelian sheaf cohomology, not just for sheaves of $\mathbf{C}$-vector spaces). This follows from properties of topological dimension of paracompact Hausdorff spaces via the "covering" definition (see Engelking's topology book) and the use of Cech theory to compute abelian sheaf cohomology on paracompact Hausdorff spaces. The crux is that locally such spaces are "proper with finite fibers" over an open ball. For $H_c$ the covering method should still work (via spectral sequence).

The only examples are central isogenous quotients of ${\rm{SL}}_1(\Delta)$ for central division algebras $\Delta$ over $k$. This is the group of norm-1 units in the unique maximal order of $\Delta$, so it is never perfect (need to argue separately depending on whether or not it accidentally is larger than the 1-units of the order).

Are you considering "split" groups (equivalently, is the Lie algebra split), or general semisimple groups (with their associated relative root systems in the sense of Borel-Tits)?

@JeskoHüttenhain: Are you reading the proof of the theorem of the cube in Mumford's book on abelian varieties? An excellent characteristic-free reference on Bertini theorems (of many flavors!) is the book by Jouanolou (just Google that name and "Bertini"). And Chow's Lemma is valid with separatedness rather than completeness (even though Hartshorne's textbook exercise imposes completeness); e.g., look in EGA II, 5.6.

@KevinVentullo: Whoops, I had not noticed the last sentence of the question. Since that endomorphism ring is $\widehat{Z}$, as noted by Strickland, for the purpose of tensoring against a torsion module it could just as well be $\mathbf{Z}$ (as I had been thinking).

@KevinVentullo: The step where you say direct limits "commute with everything" also has a small gap insofar as moving a direct limit out of the second variable of a Hom is not a generally valid thing, so its validity in the present setting (which is true) does require an argument (e.g., via Pontryagin duality or other means).

@KevinVentullo: The end of your argument has a slight "cheat": the tensor product begins as one over $\mathbf{Z}$, and one has to directly compute (by various elementary means) the endomorphism ring to see that it collapses away in the tensor product (so tensoring over the endomorphism ring is a red herring).

Yes, this is a special case of the work of Grauert-Morrey theorem on real-analytic vector bundles on real-analytic manifolds (showing that the theory is much closer to the $C^{\infty}$ case than to the complex-analytic case). Roughly speaking, real-analytic geometry turns out to be closely related to the theory of Stein spaces.

@LaurentMoret-Bailly: Good point!

The short exact sequence of interest to you consists entirely of $A/f^n A$-modules, so tensoring it against $\widehat{A}$ over $A$ is "the same" as tensoring it against $\widehat{A}/f^n \widehat{A}$ over $A/f^n A$. So you just need to check that the natural map $A/f^n A \rightarrow \widehat{A}/f^n \widehat{A}$ is an isomorphism, which follows from the fact that $f$ is not a zero-divisor in $A$. (You can just as well rename $f^n$ as $f$ for this purpose too.)