But myć się -> idź się myć, 'go and wash' is OK, right? (No z at the beginning)
zmyc sie has nothing to do with cleanliness, it's a slang expression, much like "palic gume" (burn the rubber , as the rubber of the shoe sole meaning "to get lost").
zmyc (without the reflexive sie) means "wash away" or simply "wash" as in to wash the dishes. The difference is that of the emphasis:
zmyc - remove the dirt (becoming clean is implied)
myc - to wash (removing dirt is implied)