I know at least one special case where your second question makes sense. If $G$ is a compact group, it has a category $\text{Rep}(G)$ of finite-dimensional unitary representations which break up into direct sums of irreducible representations. Fix a representation $V$ such that every irreducible representation appears in $V^{\otimes n}$ for some $n$. One can construct a graph $\Gamma(V)$ whose vertices are the irreducible representations of $G$ and where the number of edges from $A$ to $B$ is the multiplicity by which $B$ appears in $A \otimes V$. By the assumption, $\Gamma(V)$ is connected, and its combinatorial properties encode information about the behavior of the tensor powers of $V$, hence behavior about $G$.

When $G$ is finite, this graph has the property that its eigenvalues are precisely the character values $\chi_V(g)$ as $g$ runs through all conjugacy classes. But the great thing is that this statement still makes sense even when $G$ is infinite in a sense which is made precise in this blog post.

Finally, if $G$ is abelian, all of the finite-dimensional irreducible representations are one-dimensional. They can be identified with the Pontryagin dual $G^{\vee}$, which is discrete, and $\Gamma(V)$ becomes precisely the Cayley graph of $G^{\vee}$ with respect to the generators that make up $V$! So this is one sense in which the Cayley graph of an infinite group gives you algebraic data, but about its **dual** group.