Is it okay to say "Yes, we do" in answer to "Are you having a good time?"

Edit: the answer to your edited question is unambiguously “yes, we are”. Short answers use the same auxiliary as the question:

• Are you doing fine? — Yes, we are.
• Do you want some more tea? — Yes, I do.
• Will you marry me? — Yes, I will.
• Can I go to the swimming pool, daddy? — No, you can't.

It's not exactly clear what you are uncertain about, so I'll try and do my best. There is nothing specific to “having a good time in this question”. To the question “Are you playing with LEGO?”, the kids might answer “Yes, we are” (akin to your choice #1) or “Yes, with the big castle” (somewhat closer to #2), or plenty of other things. All are legitimate answers, as are the ones you propose in your question.