Translation:Please answer the question.
を is mainly for direct objects (things you do something with/to), and in the context of answering them, a question is not considered that kind of 'thing'. I think it helps to think of 質問 に こたえる as "to reply/respond to a question", which includes the directional indicator in English as well.
The correct usage is that the kanji is not used when kudasai is a helping verb (as it is in this case).
I don't think so. この is very specific as to which question to answer which is represented by "this", whereas leaving out この in this sentence makes it more general.
I can't imagine a police interrogator screaming "answer this question!", but I can imagine him yelling "answer the question!".