Do I ask the editor of the rejecting journal where to submit next?
Sometimes a referee or editor will offer an unsolicited suggestion.  When that doesn't happen, asking for suggestions would be unconventional.  As an editor, if an established professional asked me for suggestions, I'd be a little annoyed (they should be able to figure that out themselves), and I'd give a terse reply like "Sorry, I'm not sure what the best place for your paper would be."  If a beginning researcher asked me, or anyone dealing with a truly unusual situation, I would try to be more helpful.  The response you get may depend on the editor, but the worst case scenario is probably annoying the editor a little and getting no reply.  (I don't think there's any risk of dreadfully offending anyone, assuming you don't press the editor further if he/she doesn't reply or seems impatient.)  I would ask the editor only if you don't have any mentors you could ask, but in that case it could be worth a try.