Do people with angioedema have a lot of pain?

Not necesarily: Depends which part of the body is involved. Most of the cases you will feel itching and swelling. Scratching that area will lead to progressively worse swelling.

Unfortunately, yes: In fact, a common presenting feature of hereditary angioedema is a patient who has had multiple abdominal surgeries (eg appendectomy, cholecystectomy) due to history of frequent bouts of pain. Swelling below the skin and of the internal organs can be extremely painful.