It does sound old-fashioned, but it's still used, mostly as a polite form of speech in formal or professional contexts.
To avoid repetition in your third example, I would simply say:
OK, then I'll ask Tom.
Both you and the listener know you want to ask a question, so there's no need to repeat the word. The meaning is implied:
OK, then I'll ask Tom [the question].
However, the form ask a [favor/question] of is more commonly used with "favor" than with "question," which sounds even more old-fashioned.
This sentence sounds somewhat old-fashioned but very polite:
I'd like to ask a favor of Tom.
But this sentence sounds much more old-fashioned and pretentious:
I'd like to ask a question of Tom.