Learn Polish: Gdzie pani czyta? - Where do you read, madam?

In Polish, formal address requires using the third-person singular (he/she/it) form of the verb. Because you are using pani to say formal 'you' to a woman, you must use czyta (reads) instead of the informal czytasz (you read). It is exactly like asking 'Where does madam read?'.

Does this mean 'Where do you read?' or 'Where are you reading?'

It means both! Polish only has one present tense. Depending on the context, Gdzie pani czyta? can ask about a general habit ('Where do you usually read?') or something happening right now ('Where are you reading at this moment?').

Could this sentence also mean 'Where is the lady reading?'

Yes, it absolutely can. The word pani can mean a formal 'you' when speaking directly to a woman, but it also means 'the lady' or 'Ms.' when talking about someone else. You would know which one it is based on the context of your conversation.

Can I change the word order to Gdzie czyta pani?

You would be understood, but it sounds a bit unnatural. In Polish questions with a question word like gdzie, it is most common and natural to put the formal pronoun (pani or pan) immediately after the question word, followed by the verb.