Can a fall or car accident cause a protruding disc?

Yes. Disc protrusion or bulges are common in people over the age of 39/40. But following a motor vehicle accident, if you develope radiating type pain down an arm or leg, it may indicate a disc herniation with nerve root compression/ irritation. You should be checked by your pcp if these symptoms exist or if the area of pain persist for more than 2 weeks.
Possibly. Protruding discs are very common. They are sometimes painful and sometimes not. Lifting heavy objects can make disc disease worse. If there is persistent pain radiating down the legs then an evaluation should be done by a back specialist, such as a neurologist or pain specialist.