My sister says she has facial nerve paralysis. Are family members more likely to get the same thing?

Unlikely. The most commun cause of facial nerve paralysis is bells palsy. It is a unilateral paralysis of the facial nerve of unknown origin and usually the patient recovers most function within 2 to 6 months. Sometime it can leave a permanent weakness in a part of the face or eyelid. It is not familial. Nevertheless one should get a full medical exam to rule out any serious or correctable problem.
Not likely but... There are many etiologies of facial paralysis, some of which require treatment. Some of those could be genetic, but most are not. If she has not already, she needs to make sure she sees a doctor for evaluation. She may require some x-rays or other studies, and sometimes prompt medical intervention will help return her function to normal.
Facial paralysis. This is not generally an inherited condition except for a few rare conditions and can even be present at birth. In adults it is most commonly the result of bells palsy, tumors, trauma/ or injury.
No. Although there are some syndromes associated with facial paralysis like melkersson-rosenthal syndrome, these don't run in families. It is not contagious.