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A few more details regarding the last sentence. If $G$ is a countable group and $\tau \in \mathcal{T}^0_G$ is a complete extension of $A_G$, then $\tau$ corresponds to a ring with underlying additive group $G$ if and only if there is no single formula $\phi(v_1)$ such that $\tau \vdash \phi(v_1) \to v_1 \neq a$ for every $a \in G$. Of course, this happens exactly when $(\forall v_1 \lnot\phi(v_1)) \in \tau$ or $\phi(a) \in \tau$ for some $a \in G$. So the points of $\mathcal{T}^0_G$ that correspond to types that contain one of these for each formula $\phi(v_1)$. Since $G$ is countable, this is always a $G_\delta$ subset of $\mathcal{T}^0_G$ and therefore it is always a nice Polish subspace of $\mathcal{T}^0_G$.

Since you're interested in the action of $\mathrm{Aut}(G)$, let me add that $\mathrm{Aut}(G)$ acts on $\mathcal{T}^0_G$ in a natural way and that the $G_\delta$ subset described above is invariant under this action. However, since $\mathcal{T}^0_G$ is compact, there may be much to gain in considering the action on the whole space.
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Being a logician, I would look at this from the model theoretic point of view and think about spaces of types, which are all rather nice Stone spaces.

Here is the setup for your case for $0$-types. Let $\mathcal{L}_G$ be the language of rings augmented with a constant for each element of the additive group $G$. Let $A_G$ be the set of purely additive (have no mention of multiplication or the multiplicative identity) sentences of $\mathcal{L}_G$ that are true in $G$. This is a partial $0$-type and the space $\mathcal{T}_G^0$ of all (complete) $0$-types extending $A_G$ is a lot like what you describe. Indeed, if $a,b,c \in G$ then $a\times b = c$ is a sentence of $\mathcal{L}_G$ which determines a basic clopen set $\lbrace p \in \mathcal{T}_G^0 : a \times b = c \in p\rbrace$. This is a little finer than the space $\mathcal{R}_G$ you describe since elements of $\mathcal{R}_G$ only encode the truth for quantifier-free sentences of $\mathcal{L}_G$. The advantage of using spaces of types is that they have been extensively studied in model theory. Beyond $0$-types, you can consider the spaces of $k$-types, which instead of sentences use the broader class of formulas of $\mathcal{L}_G$ with free variables among $v_1,\dots,v_k$. Together, these spaces of types give a very nice understanding of the situation you're looking at.

Note that there are some types that are not necessarily compatible with the underlying additive group being precisely $G$. For example, the $1$-types that extend $\lbrace v_1 \neq a : a \in G \rbrace$ are incompatible with such rings. Nevertheless, if $R$ is a ring with underlying additive group $G$, then $R$ corresponds to the $0$-type $\lbrace \phi : R \vDash \phi\rbrace$, every element $a \in R$ corresponds to the $1$-type $\lbrace \phi(v_1) : R \vDash \phi(a)\rbrace$, and so on. The Omitting Types Theorem lets you know which types can correspond to rings whose underlying groups are precisely $G$.