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Oh, I see, I should have looked at your example more carefully. (I also see now that you want associative unital algebras, which is also not what I had in mind.) I think that your definition of "compatible" is less natural (because the elements of $G$ do not induce automorphisms of this algebra structure), but of course your question makes perfect sense.
@YCor: I think the OP means neither of the two, but just the set $\mathbb{Z}/n\mathbb{Z}$ with the action given by multiplication by an invertible element $k$ modulo $n$.
Did you read the "What is ... a Leavitt path algebra?" (by Gene Abrams) in the AMS notices? See ams.org/publications/journals/notices/201608/rnoti-p910.pdf. It explains to some extent why the defining relations are natural from the point of view of the Invariant Basis Number property of rings.
Somehow, prime powers and roots of unity feel "dual" to each other. Perhaps this can be made precise through the fact that as locally compact abelian groups, $\mathbb{Z}$ and $\mathbb{S}^1$ are dual to each other, but don't ask me how to relate this fact to the examples that you mention...
About your Note 2: I would guess that your friend was thinking in terms of affine group schemes, in which case he might have been thinking e.g. about the group $\mu_p$ of $p$-th roots of unity in characteristic $p$.
If I understand the definition correctly, these "maraos" are precisely what people in finite geometry call "$k$-spreads". They have been studied quite a bit, not only in projective spaces but also in polar spaces.
Kac-Moody algebras in general are constructed from an arbitrary generalized Cartan matrix, which you could think of as a generalization of an ordinary Dynkin diagram. Is this the sort of idea you have in mind?
