Why didn't she ask herself that question?
The two answers on the multiple chose question were "Warum hat sie sich diese Frage nicht gestellt?" and "Warum hat sie sich die Frage nicht gestellt?". Why is the second one acceptable? How does "die" translate to "that"?
Old English sē, sēo, þæt (masc/fem/neut) split up into a definite article "the" and a demonstrative "that" -- originally, the word could be either.
German has kept this old behaviour, and der, die, das can be either a definite article (like English "the") or a demonstrative determiner (like English "that").
So the German word for "that" (or "this" -- German doesn't make as strong a distinction here as English does) can be not only das but also der or die or dem or den or des, depending on the gender, number, and case of the noun that follows.
Both questions mean in German about the same but "diese" is more emphasizing this word. I say "die" if the person, I'm speaking to, knows which question I mean or if I think he or she knows it. If I'm not sure whether one knows it I can use "die" as a reflexive pronoun: 'die Frage, über die wir hier reden,' (the question we are talking about).
No - Warum hat sie selber die Frage nicht gestellt? would mean "Why did not she herself ask the question?" or "Why did she not ask the question herself?", i.e. asking why the asker was not "she herself [rather than someone else]". Similarly with Warum hat sie die Frage nicht selber gestellt? which would also mean "Why didn't she ask the question herself [instead of requesting that someone else do so for her]?"