In the case of a hypersurface of dimension $d$, or any Gorenstein singularity of dimension $d$, $E \cong H^d_{\mathfrak{m}}(R)$ (of course, this isomorphism is up to multiplication by a unit). $G$ should act on $H^d_{\mathfrak{m}}(R)$ directly (you should even be able to do this explicitly via Cech cohomology). This should be enough for question 1 in your setting.

For question 2, I'm pretty sure the answer is no in general. In particular, the socle of $E$ as an $R^G$-module, is probably not 1-dimensional. Note the socle of the injective hull of the residue field is always 1-dimensional (see for example Bruns and Herzog's book).

Let me give an example. Suppose $R = \mathbb{C}[x,y]$, $\mathfrak{m}$ is the origin, and $G = \mathbb{Z}/2$ acts on $R$ by multiplying the variables by ${-1}$. Then $R^G = \mathbb{C}[x^2, xy, y^2]$. The socle (elements killed by the maximal ideal) of $H^2_{\mathfrak{m}}(R)$ as an $R$-module is just the Cech class $[1/(xy)]$. The socle as an $R^G$-module however also includes the elements $[1/(x^2y)], [1/(xy^2)], [1/(x^2y^2)]$ since the relevant maximal ideal of $R^G$ is $(x^2, xy, y^2)$.

The point is that the socle as an $R^G$-module is all elements killed by $x^2, xy$ and $y^2$.