What are some of the symptoms of a broken rotator cup in the shoulder?

Pain . The actual term is a "cuff", and it describes tendons that wrap around the top of your humerus at your shoulder, like the end of your sleeve wraps around your wrist. Because it is soft tissue, it doesn't "break", but can tear. Pain and decreased movement are the most common symptoms of arotator cuff tear, but are also symptoms of many other shoulder problems.
Depends... Rotator cuff tears are very common. They are found in many people who have no pain at all. Some tears are caused by trauma or a fall, and some are attritional ---where the tendon simply wears out. Not all rotator cuff tears hurt, but some do. Torn rotator cuffs will also lead to weakness if they are large. Not all rotator cuff tears require treatment, only some do. Talk to your doctor.