Learn Polish: Gdzie pan mieszka? - Where do you live, sir?

In Polish, when you speak formally to a man, you use pan (sir or Mr.). Even though you mean "you", the word pan is treated grammatically like "he". That is why we use the third-person verb mieszka ("he lives") instead of the informal "you" form mieszkasz.

How would I ask this same question formally to a woman?

You would replace pan with the formal word for a woman, pani. The verb stays exactly the same because pani also takes the third-person form. The sentence becomes Gdzie pani mieszka?

If someone asks me this, how do I use a negative to say "I don't live in Poland"?

To make a negative sentence in Polish, you simply place the word nie immediately before the verb. You would answer Nie mieszkam w Polsce.

Is there a Polish equivalent to the word "do" in "Where do you live?"

No, Polish does not use helper verbs like "do" or "does" to form questions. You simply use the question word gdzie followed by the subject and the main verb.