I would say that they are not interchangeable, at least on nuance. Sentence 1 would normally be used by someone who would prefer that he call more often, he doesn't call often enough. The other two sentences would normally be used by someone with the opposite complaint, his calls are too frequent.

Yes, they can be used interchangeably. But "he calls me on a whim" is slightly different from the others (more positive). You tend to do something on a whim (i.e. spontaneously) when it brings you joy. Depending on context, "whenever he likes" and "whenever he pleases" can also be positive, but tend to usually be used in a negative context.

The first three can be compliants and mean more or less the same thing. The last one, “on a whim,“ could actually be said, for example, by a mother who is happy that her son sometimes just picks up the phone and calls her to say hello

My first impression is that this is what a woman is saying about her boyfriend. In this context they all seem similar, but all three interpretations could hinge on how the sentences are said, and where the stress lies. If I could see videos of the three sentences with a woman in varying stages of anger, that would help.