Yes, you need an article. Without it, I hear a name: Jim. "Are you going to the gym tomorrow?" We use the specific "the" because we assume the person will go to the same gym that they usually go to. If you write, "are you going to a gym tomorrow?", it means that neither of you know which gym, plus that this may be their first ever visit to a gym.

Well, both are possible. If you and I are friends and I know that you go to the gym on a regular basis, I could ask you. Are you going to the gym? Since I would most likely know the gym that you are going to. But if I know you and I have never seen you going to a gym or doing exercises, and I see you on the street dressed in such a way it would make me think that you are going to the gym, but I have no idea what gym it could be, then I would say "are you going to a gym?".

Yes, people usually say "the gym". It's the same as people referring to the airport or the hospital, even when it might not be clear to which exact place they are referring.

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I often hear people say "are you going to *the* gym tomorrow?"