Learn Dutch: Waarom verleent hij de gunst aan de buurvrouw? - Why does he grant the favor to the neighbor?

Yes, but you would have to change the word order. If you remove the preposition aan, the indirect object (de buurvrouw) must be placed before the definite direct object (de gunst). The sentence would then become: Waarom verleent hij de buurvrouw de gunst?

Is aan always the right preposition to use with verlenen?

Yes, the standard expression is een gunst verlenen aan (to grant a favor to). The preposition aan (to) is the default choice in Dutch to indicate the receiver of an action or object, which perfectly fits the act of granting something to someone.

Why is the verb verleent placed before the subject hij?

Because this is an open question starting with the question word Waarom (why). This triggers inversion in Dutch, meaning the conjugated verb must take the second position right after the question word, and the subject (hij) must immediately follow it.

Does buurvrouw mean any neighbor, or specifically a female one?

It specifically means a female neighbor. The word is a combination of buur (neighbor) and vrouw (woman). The male equivalent is buurman. If you want to talk about neighbors in general, you usually just use the plural form buren.