$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced by the adjoint action. Assume $|W|$ is invertible in $k$, so Chevalley-Todd-Shepard implies the quotient is affine space.

I expect the following basic geometric properties to generalize from the $\operatorname{GL}_n$ case.

- The $k$-points of $\g/\!/G$ are in bijection with Zariski-closed $k$-orbits.
- Every $k$-rational orbit has a unique closed orbit in its closure, and the map on $k$-points is this identification.
- The locus of regular elements (those $X \in \g$ with $\dim G_X = \operatorname{rank} G$, as small as possible) is exactly the smooth locus of the morphism.
- The semisimple $X \in \g$ (in the sense of additive Jordan-Chevalley) are exactly those with closed $k$-orbits.

Do these hold in the setting above? If so, how much more general are they? The first two at least seem like general facts about actions on an affine $k$-scheme.

**Edit**: After some more reading and Ben's helpful answer, I was able to resolve most of this, though I haven't found anything that ensures **surjectivity of the quotient morphism** in the positive characteristic case: in characteristic zero the argument uses that for any $A^G$-algebra $S$, we have the identity $S = (S \otimes_{A^G} A)^G$. This doesn't work in positive characteristic (and *GIT* never claims that $(\operatorname{Spec} A)/\!/G$ is a "universal categorical quotient" unless char 0).

I'll leave this open for a bit in case Ben or anyone else can chip in with a substitute argument or counterexample.