Why do we associate a graph to a ring?

The following answer basically involves things I learned about from others in a conversation on the topic of this question, which I have heard voiced many times and is a reasonable question. The paper ANISOTROPIC GROUPS OF TYPE $A_n$ AND THE COMMUTING GRAPH OF FINITE SIMPLE GROUPS by Yoav Segev and Gary M. Seitz uses the commuting graph of a finite simple group to make progress on the Margulis-Platanov conjecture. This earlier Annals paper of Segev On finite homomorphic images of the multiplicative group of a division algebra also uses commuting graphs to solve known conjectures. Of course the graph is not the only thing used and I am not enough of an expert on the subject to say how central the graphs are to the paper.

So the commuting graph of a finite group definitely came up naturally. I am unaware of similar ring theoretic examples. I also am unaware of how planarity and other graph theoretic properties of such graphs play a role in any applications.

Let me also note the nice paper of Peter Cameron The power graph of a finite group, II which shows that groups with isomorphic power graphs have the same number of elements of each order. This wasn't motivated from outside the theory but I think is still a sign the graph is relevant.

I think some aspect of the answer is still missing that is not shared by other topicA-vs-topicB-type-questions: Graphs can (sometimes) be drawn. Even if no graph theoretic technique ever solves a ring theoretic question and no ring theory ever improves our understanding of graph theory, the fact that we can draw or otherwise visualise certain graphs can be a big boost to understanding in and of itself.

Maybe there is some structure in the ring in question that is purely ring theoretical, useful for the problem at hand, but somewhat difficult to discover. The very fact that one can draw such a structure as a graph makes it visible and intuitively graspable. Human brains can find visual patterns in a fraction of a second. Finding patterns in complex algebraic structures is (many) order of magnitude slower and harder for human brains. This alone can be huge a benefit of associating graphs to algebraic (or other non-trivial) objects.

And there does not need to be any interaction between ring and graph theory for this benefit. The very fact that one can do an inductive but purely algebraic argument, say by induction over the vertices of the graph, is often enough. You just had to draw the graph to see what the right ordering for the induction is (maybe you inductively delete leaves from a tree or something common like that), nothing more.

Your question (2) seems to me a completely valid question. I'm not aware of any old questions solved by the graphs you mention in your question, and I'd be interested to hear of examples, especially for graphs associated to rings. But there is one prominent example of a graph construction that has been used to solve many questions about groups.

The Cayley graph of a finitely generated group $G$ carries a well-defined metric, the word metric. Up to quasi-isometry, the resulting metric space is an invariant of $G$, and the whole field of geometric group theory is concerned with relating the algebraic structure of $G$ to the metric structure of its Cayley graph.

Perhaps the most classical example of a group-theory problem that was solved using these techniques is the Burnside problem. (Admittedly, the first solutions were not geometric, but geometric techniques have led to solutions for the best known exponents.) One could give many other examples -- the fact that a random finitely presented group is infinite and torsion-free springs to mind.

ADDED:

In fact, what makes Cayley graphs so useful is that they carry a natural action of $G$. Another strand of research in geometric group theory studies automorphism groups $\mathrm{Aut}(G)$ via their actions on graphs constructed from algebraic features of $G$. Bill Harvey's curve graph of a surface, and various graphs associated to the outer space of a free group, are perhaps the most prominent examples.

In principle, the commuting graphs mentioned in the question could be used for this kind of purpose. I'd be interested to hear about instances where they have been.