Is my shoulder weak because it produces a popping sound?

Not necessarily. The popping sound in a shoulder is called crepitus. It usually results from some scar tissue developing in the shoulder. Generally around the age of 35 or above the shoulder starts to make noise. It doesn't mean anything is wrong but rather that the shoulder is not as lubricated as it once was and that some scar tissue has developed.
Physical exam needed. The answer to this question depends upon physical examination findings of your shoulder. The popping sound can come from a myriad of sources, to include, but not limited to: scar tissue, tendons changing position, a torn rotator cuff edge, a torn labrum within the joint, and a loose body or flap of cartilage (joint lining). Only an appropriate physical examination will discern the true cause.