Should a student shake hands with the professor?

Greet them in whatever way is natural to you. In university settings (where contact with different cultures is common, more so in graduate studies) they'll probably take it in stride. If they offer to shake hands, do so. Doing otherwise is certainly rude.

In the U.S., it is acceptable to shake hands in this environment. But in the classes that I teach, it normally only happens after the last lecture to say farewell. The handshake isn't required. Some students have kept a distance, while politely thanking me for teaching the class. That is perfectly acceptable too. Other students have thanked me via email. It depends on what the student is comfortable with.