Have I got a mental disorder?

You appear to be having some very strong symptoms of OCD (Obsessive Compulsive Disorder) which oftens responds great to treatment. The anxiety disorder which you are also exhibiting is definitely connected to the OCD (although it probably has other behavioral components as well) and is very treatable. Visit your primary physician and ask for a referral to a good psychiatrist and psychologist. Don't let fear get in your way! This is no way to live!