Translation:Why didn't she ask herself that question.
Yes - that means something else.
Why didn't she ask herself this question? = Why did she not ask this question to herself?
Why didn't she herself ask this question? = Why did not she (and nobody else) ask this question (to someone)?
(The translation for that is the same as for Why didn't she ask that question herself?; see the thread started by paulcan.)
No, it does not. Your sentence would be Warum hat sie die Frage nicht selbst gestellt?.
Duo’s sentence is about asking yourself (i.e. you yourself are the listener; you are speaking to yourself when you ask the question), while your sentence is about asking yourself (as opposed to someone else asking) and leaves open who the listener is.