Addressing a professor - culture clash

A lot of us who live in both worlds (I am German and frequently visit there, but live and work in the US), we call colleagues we don't know well by their first name when we meet in the US, and by Professor X when we meet in Germany. In other words, we use the common convention of the place we're in. That's probably a good rule for you to live by as well.

By default, write Prof. Dr. Müller (or whatever his name is) and say Professor Müller at least in German. This is the golden standard. Addressing him in English may but need not be different from the above. In any case, if you start with the golden standard above in any of the two languages, you do nothing wrong, and, after that, it would be up to him to suggest a different form.

If you really wish to touch this topic first, a good manner would be "Wie möchten Sie am liebsten von mir angeredet und angeschrieben werden?" (“How would you like me to address you orally and in writing?”)

You are certainly overthinking it. Your professor has foreign experience, so he knows that there are different cultures. If he would care about your behaviour, he would signal it to you, e.g. resulting in a dialogue of the form:

You: "Peter, hast Du ... ?"

Professor: "Nein, Herr Schmidt, Sie muessen noch ...".

If you are in a situation where you expect that the use of Du/Sie and first name is important, you can use passive constructions to avoid explicit addresses. Then the other person can decide how to response, and you response in the same way.

Another common technique is the use of "Ihr", in particular in shops, clubs, or other groups. Its literal meaning is informal plural address, but it can be understood as addressing the person you talk to as well as the shop as a whole. Using "Ihr" in such a situation signals that you are comfortable with "Du" without forcing your counterpart. However, "Ihr" should not be used to a person which cannot be identified with a group, because in such a context it would appear extremely quaint.