Do lots of people get rotator cuff syndrome, or is it unusual?

Hm. Complicated. I'm not sure there is a specific "syndrome" but plenty of ppl, especially the eldery and younger active overhand throwing athletes, show damage to the rotator cuff. A portion of these people will be symptomatic from the damage. Acute ruptures in young people may be repaired surgically. In the elderly they may be treated with phys therapy only.
Impingement syndrome. In my practice, which is limited to disorders of the shoulder and elbow, problems related to the rotator cuff are the most common complaints i see.
Not unusual. Rotator cuff syndrome/strain/tendinitis is extremely common. Usually first treated with anti-inflammatory medications and physical therapy.
Not that common. Rotator cuff tear , really cause inability to move the arm up, not just pain--more common inflamation of the tendons called tendinitis--get an exam.