"Did you go to the cinema?" ...when? Past simple requires the time of the action. For example, "Did you go to the cinema last week?" "Have you been to the cinema?" covers any time up to now.

Both are correct. I agree with Dan the phrase using "been" feels more like a question about the experience at the cinema whereas the one with "go" is more about the act of going to the building as possibly one in a sequence of activities. I'm British and the context would dictate which form I chose. So I would ask the question "Have you been to the bank?" in order to get confirmation that the person had gone and encourage expansion on what happened whilst they were there. If I asked "Did you go to the bank?" I would probably be expecting just a yes or no.

You can use either. There's almost no difference in meaning. The difference is so small that it's hard to explain. This is what I think, but maybe other people will give different answers. "Have you been..." focusses attention on the completion of the event. "Did you go..." focusses attention on the start of the event, or on the event as a whole. It's really subtle. (I'm a U.S. speaker and we say "movies," not "cinema" but I'll use "cinema" anyway). If someone asks me "Have you been to the cinema" I'd be more likely to answer something like "Yes, and then we did some shopping at the mall." If someone asks me "Did you go to the cinema," I'd be more likely to say something like "Yes, we saw 'The Imitation Game.'" I have an impression British speakers would be more likely to say "Have you been" and U.S. speakers would be more likely to say "Did you go" but I'm really not sure.