How dangerous are dog bites on the hand?

Can be dangerous. Any bite to the hand from dog, cat or human can be dangerous and should be cleaned and possibly explored surgically. There are a large number of very important structures in the hand as well as a risk of spread of infection that can be devastating if untreated. If in doubt have your doctor evaluate it.
Depends. Dog bites can easily become infected, and hand infections can lead to significant dysfunction. Get treated immediately.