You're absolutely right that there are two distinct meanings, and that your 1. (meaning "request") does not usually take a direct object, i.e. it usually requires "for", whereas 2 (meaning "put a question") does take a direct object.
Sense 2. can take other words meaning "question", eg "ask a conundrum", or it can take a clause representing the question as a direct object, eg "ask whether this is right".
One exception I can think of to the "usually" above, is that you can ask "a favour" and words of similar meaning - permission, leave, a boon, someone's indulgence. You can "ask" these or "ask for" them.
I'm not sure how much you can make use of the special English construction of putting the indirect object before the direct, without a preposition, with these words:
He asked me a favour.
sounds as good to me as
He asked me a question.
But I'm not sure about
?He asked me permission to go.
I think I would prefer
He asked me for permission to go.