How far apart are these descriptions? I was approaching the issue from the perspective of erotetic logic, and my intuition is that self-evidence is when a proposition is evident from its erotetic format ("S is P," is self-evident iff the question, "Is S P?" is evidence for, "S is P"), but being self-explanatory is subtly different. I.e. instead this is: "S is P," is self-explanatory iff our ability to ask the question, "Is S P?" is what explains how S is P.
This seems as if it would restrict self-evidence and self-explanation to pure erotetic relations, maybe. Like, only Cartesian ur-sentences (about the faculty of doubt) would be self-evident or self-explanatory.
The last disjunct of my question(!), then, is whether there are self-justifying imperatives and self-questioning questions. "Why do the right thing?" would seem to self-justify the imperative, "Do the right thing," in the sense that it's unreasonable to ask what the right thing to do is unless you care enough to act on that knowledge. As for the last category, this would be things like:
Is this a question? / Q: Is Q a question?
What is the first question on the Test?
(Although that last of the last things, here, is a riddle "too"...)