Common. Many patients react to different procedures like you do. It's common for patients to feel smothered with impressions, root canal with rubber dams, and procedures where access is difficult. In the event that you really need these procedures but just won't see a dentist for necessary treatment, i suggest you research sedation dentistry. Dentists will work with you to overcome your anxiety.
It's possible. Working in the mouth can be very tight. You have the hands and instruments of both the doctor and the assistant. If you are the type of person who feels crowded when folks are too close or has a small mouth, it might be a good idea to advise the doctor or assistant in advance so they can plan for breaks in the treatment to allow you "catch your breath". Just maintain good communication!