WRIST RASH. One common transferable disease that can occur on the wrist is a ringworm (fungal) infection. Are you around any cats or dogs? Another can be impetigo. Do you wrestle? See a dermatologist if it doesn't disappear in a few days. Without actually seeing your rash it's impossible to k now.
Probably not. You probably have a contact dermatitis from your watch band or some wrist jewelry would be my guess. Let your doc take a look at it and see if they can help.
Yes. We worry about MRSA (methicillin-resistant staphylococcus aureus) as well as fungal infections predominantly.