Replacing Rubber Gasket

The house is five years old. The tub is on the second floor, so I know we'd be able to see a major leak if it happened. I am more concerned about the potential of a minor leak that is not as obvious. I have no clue why the gasket would be worn so much. If the black is not the gasket, I am not sure what else it could be. I can scrub it away, but as soon as we Tilex the shower, the bleach accumulates around the drain and the black comes out from the drain; however, the black will come out even when we immediately wash the bleach away. We also have the black ring in one of the sinks. I am not sure if it could be the metal, soap scum or if it is the gasket. I know it is difficult to diagnose without actually being there. We really don't know what the next step is. I want to hire a plumber, but we really are financially tight. So, a gasket in good shape shouldn't break down with bleach? What would you say our next step is? Is there any way to find out if there is a minor leak? Oh, and thanks, Cass, so much for your help!