Yes, that would be fine. English has two ways of expressing this, with a slight nuance between them, but Esperanto only has one.
At least, most speakers of Esperanto wouldn't usually bother to make the distinction that English people might choose to naturally make between "He didn't answer" and "He hasn't answered". However, if it's really important, you can always resort to "Li estas ne respondinta al mia demando". But frankly, nobody does!
No, it has to do with whether the verb is transitive or intransitive, and whether the noun phrase in question is the direct object or the indirect object.
In this case, "respondi" is transitive, just as it is in English:
Li ne respondis mian demandon.
He did not answer my question.
But consider "to respond". It's intransitive:
He did not respond to my question.
For "Give me the book", "the book" is the direct object of "give" (it's the thing that is given) and "me" is the indirect object (it's the recipient of the direct object). In English, we can say either "Give me the book" or "Give the book to me".
daniel, I thought Rae.F answered what you were confused about, so I'm not sure if I understand...?
"Respond to" is not exactly "react to"
In the examples Rae.F gave, "respond" is intransitive while "answer" is transitive. But I believe (correct me if I'm wrong) that "answer" can be used in either a transitive or intransitive manner. Likewise, respondis seems to be flexible in that way, according to Duolingo, which accepts either answer for the translation question regarding this sentence.