If someone asks you what are you doing, can you answer what you are being done to?
Bob: What are you doing?
Joe: I am being licked by my dog.
Being is a verb, like is and are. Can I say the answer is logical, or "correct"?
You can answer this question with any gerund (-ing), passive or active.
Your example is very good.
Yes, Joe's answer is logical and a natural way of expressing that his dog is licking him. It might help to think of it as describing a state of being.
Yes, that is correct. You can say:
My dog is licking me.
Or, as in your example, you can say:
I am being licked by my dog.
Both ways are correct. The first one is in the "active voice". The second one is in the "passive voice".
The difference is that the first one puts the emphasis on the dog, whereas the second one puts the emphasis on "me" (or "I" in the sentence).
You can look up "active voice" and "passive voice" if you want to know more.