"Please answer the question."
I think it's because you're not performing an action ON the "question" (where it would be a direct object of an action and would require WO particle) and instead the "question" is the TARGET of the action.
Since I started thinking about the NI particle as a "particle that points to a target of the action" I stopped having trouble with it.
That's why it will be 質問に答える and 質問を書く or 質問を忘れる, because in the first the question is the target, while in the last two you are doing something directly to the question. Similarily, the people are a TARGET when you talk to them (彼に話す) and a DIRECT OBJECT when you hit them. It's also why you will use 私には娘がいます when saying that you have a daughter, literal translation will be: "As for I (<--target) daughter exists". That's why you use NI with a date only when it's a specific point in time (when that date is a TARGET for the action), and it's the same with the locations.
A lot of it has to do with context, I think. The language is structured completely differently than English, with all kinds of nuances that take time to learn. That being said, if you think something should be accepted, report it. Remember, it's still in beta.
My understanding is that 質問 is a question that would come up in a conversation. An "inquiry" if you will. More relaxed. 問題 on the other hand is a question that is more interrogative. A problem on an exam, or a discussion of social issues. More serious. This is only off my own learning however and I'm I'm open to correction.