I understand that when individuals have single-peaked preferences, majority rule will be transitive and the choice set will be non-empty. So single peaked preferences is a sufficient condition.
Is it also necessary? If not, what other domain restriction can given majority rule a non empty choice set?
No, single-peakedness is not necessary for majority rule to be transitive. For example, "single-dipped" preference profiles (the vertical mirror image of "single-peaked" profiles) also produce transitive majority preferences. A much-more general condition is "triple-wise value restriction", which Sen (1966) showed generates a transitive majority preference relation. Here is a good summary of these and related concepts, with some references to the primary literature:
https://plato.stanford.edu/entries/social-choice/#NonDicPreAggRul
