@dima: Determinants of that form are Schur polynomials times the Vandermonde determinant.

The Grassmannian $G(k,n)$ is toric if and only if $k = 1$ or $k = n - 1$, in which case it is isomorphic to $\mathbb{P}^{n-1}$. Instead of having a dense orbit for a torus in general, it always has a dense orbit for a Borel subgroup of $SL(n)$ (e.g. the subgroup of upper triangular matrices of determinant 1).

In fact, in the construction of a Mori Dream Space $X$ as a GIT quotient $V // G$ via the Cox ring, $V$ is an affine space if and only if $X$ is a toric variety.

Wow, Nate Silver is everywhere these days. (On a slightly more serious note, it's nice to give first and last name for someone who you cite, in case a curious novice is interested in searching for the individual to learn more about his work.)

You are right that the singularities of $\overline{X}$ will likely play an important role. I believe there are versions of the localization theorem that apply to singular varieties, or alternatively you may try to find a $T$-equivariant resolution of $\overline{X}$. You might look at Brion's survey article "Equivariant Cohomology and Equivariant Intersection Theory", which has several versions of the localization theorem (and references to other work as well).

Assuming that the action of $T$ on $X \subset \mathbb{P}(V)$ is induced from an action of $T$ on $V$, a natural thing to do would be to study $H_T(\overline{X})$ (or possibly some related complete variety), where $\overline{X}$ is the Zariski closure of $X$, using localization and then use that to recover information on $H_T(X)$. Your answer is going to depend in some way on what is going on "at infinity".

I am generally considering an algebraically closed field, and I am fine with characteristic zero assumptions if necessary. I don't think reductive adds anything important, but I tried to cast as wide a net as possible while fishing for examples. I'll add an edit to say a little about motivation.

The assertion formerly known as Theorem 2? With some LaTeX work, you could then use the Prince symbol to refer to it in the paper. en.wikipedia.org/wiki/…

The idea of the projective plane is at least implicit in the work of the ancient Greek Pappus. It's not clear to me that one can say when projective space was developed. Anyway, it's still an interesting question, but I think it might be helpful to indicate exactly which aspect of projective space you are interested in. (I realize your second question does that, but I didn't know how to interpret your first question. Are you specifically interested in when people understood the generalization to $n$-dimensional space, or the first incarnations when $n = 2, 3$?)

Start here? en.wikipedia.org/wiki/Projective_geometry