Answering questions is a pretty important part of learning, and if we don't answer questions we don't continue to evolve. That doesn't mean the question has to be answered right now, it just means we should answer the question eventually. Of course, this is talking about all of the collective questions humanity has as a whole, not literally every single question ever asked.
Yes, I think that every question needs to be answered, because that is the only way that we can learn and move society forward. Everything that we have today is because someone asked a question, and wanted to find out the question so we found out more than we previously knew.
Questions do not always need to be answered because some do not require a reply. For example, many people ask questions in a joking format or already know the answer to the question they are asking. Redundancies and jokes should be removed, and then all questions could be answered outside of those.
It is not possible to answer every question. While most should be, some simply do not have an answer. We don't know the answer, and it's better to just say so than to say something that isn't true and have people rely on your untrue answer for years. Some questions are rhetorical and the person asking never expected to get an answer.
No, we don't need to answer every question that is asked. And contrary to popular belief, there are, in fact, stupid questions. Very stupid questions. There are also dangerous and silly questions that do not deserve to be pondered by anything. Curiosity isn't always a good thing, and this goes especially for questions.
No, I don't think that questions always have to be answered. I think that there are many questions like those of the rhetorical nature that are posed and not expected to have a concrete answer. There are also very many important questions like the origin of life that will probably remain mysterious for a long time. We might have a lot of evidence, but getting a solid answer is probably not going to be very easy or possible at all.