In the dictionary it says paranoia is a mental condition. People say that Skipper is paranoid A LOT. Does this mean Skipper is…mental?

No. While paranoia is indeed a mental disorder, the term is often used to refer to just "baseless or excessive suspicion of the motives of others" (Dictionary.com). Skipper's paranoia fits this segundo definition.

Lady I can assure you with every bone in my body that I am NOT mental. Not even close.