Probably, though I suppose it's possible that there are multiple crystal structures and I want a different one. Anyway I want the unraisable ones.

Well, you can see the paper: front.math.ucdavis.edu/1503.04070

Well that was straightforward! I'm hoping someone has a more positive answer (i.e. extra conditions to require to make it possible), so I won't accept this quite yet, but it's certainly a definitive answer to the question as stated.

I don't know that he wrote it anywhere (but I won't say he didn't); he told me about it in person at some point.

Yes, but the transversality results in differential topology involve making generic choices in an infinite-dimensional space rather than a finite-dimensional one.

The famous open problem is whether the ideal generated by those $n^2$ quadratic equations $(AB-BA)_{ij} = 0$ is radical.

Demazure has a paper about the automorphism group of toric varieties, which can be nonreductive, e.g. $\mathbb P^2$ blown up at a point. Then I guess your question is about maximal tori of that automorphism group? They're still all conjugate as I recall, i.e., the resulting toric varieties are equivariantly isomorphic. Things change a great deal if you consider different toric variety structures on the same real manifold, e.g., all the even Hirzebruch surfaces are toric and complexly different but really diffeomorphic.

I'm a little surprised that Kirwan's thesis "Topology of quotients in symplectic and algebraic geometry" hasn't been mentioned yet, as a place to see GIT in a rather differential-geometric perspective. Of course, Kirwan is coauthor on the 3rd edition of GIT.