Is wrist pain common after shoulder surgery?

Yes. After shoulder surgery, any swelling that occurs will typically travel down and down and down. It ends up around the elbow and the wrist. Because of this, the outside of the elbow and top of wrist both often hurt after shoulder procedures such as shoulder arthroscopy or shoulder replacement.
Not really. I have seen this in a few of my patients. I use an arm holder during surgery. Usually the operative arm is strapped into this. Once in a while, patients will have some pain from having the arm in a certain position for a prolonged period of time. I probably see this 1-2 out of 300 procedures per year.
Wrist pain. No it is not too common. Some reasons: related to swelling in arm, use of an arm holder during the surgery causing wrist pressure, or improper sling wear.