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.
Please tell me if it is right. Can もんだい be used for both problem and question while しつもん only means question ?
Mostly- an exam "question" is essentially a "problem", which both words can be used for. But no, you would not use しつもん to refer to a problem you're having with erroneous Japanese language courses, for example :)
質問 (question) に答えて (answer) 下さい (please...). I don't think the kanji for ください is often written though.
The correct usage is that the kanji is not used when kudasai is a helping verb (as it is in this case).
Thank you, I changed the link to a working one. Correct grammatical usage is to write in kana, though many native speakers use the kanji anyway.
Even though there's no この, in English "answer this question" and "answer the question" (what I put) are essentially the same.
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!".
For the third time today, duolingo is marking an answer wrong that is the exact same as the "correct" answer. This only happens in Japanese, never in French or Russian. Get it together, duolingo Japanese team.