The answer to the first question is yes. Although the words "equivariant cohomology" don't appear there, this goes back to Deligne's Hodge III, since he defines the mixed Hodge structure on a simplicial variety and you can define the Borel construction simplicially in the usual way (via the resolution with $X$, $G \times X$, $G \times G \times X$, and so on). In particular he proves that $H^\bullet(BG)$ is pure of Tate type. If $G$ is connected, then the spectral sequence $$H^p(BG) \otimes H^q(X) \implies H^{p+q}_G(X)$$ must therefore degenerate immediately for weight reasons and so the answer to your second question is also yes.

I would imagine that there is a suitable theory of "six functors" for simplicial varieties as well, but I don't know the literature well enough to say anything meaningful or give a reference.

The answer to the first question is yes. Although the words "equivariant cohomology" don't appear there, this goes back to Deligne's Hodge III, since he defines the mixed Hodge structure on a simplicial variety and you can define the Borel construction simplicially in the usual way (via the resolution with $X$, $G \times X$, $G \times G \times X$, and so on). In particular he proves that $H^\bullet(BG)$ is pure of Tate type. If $G$ is connected, then there is a spectral sequence $$H^p(BG) \otimes H^q(X) \implies H^{p+q}_G(X).$$ If $X$ is smooth and projective this must therefore degenerate immediately for weight reasons and so the equivariant cohomology of smooth projective varieties is very simple: $H^\bullet_G(X) \cong H^\bullet(X) \otimes H^\bullet(BG)$. In particular the answer to your second question is also yes.

I would imagine that there is a suitable theory of "six functors" for simplicial varieties as well, but I don't know the literature well enough to say anything meaningful or give a reference.