Refrigerator was hot to touch so cleaned coils but was still getting hot. Pulled off cardboard from rear of frig and pulled out a dead mouse which made the fan come on but not it's rattling and too loud. Now what?

Perhaps the fan burnt out while the mouse was inside of the fan, did the mouse stop the fan from moving? Maybe the screws/bolts are loose?

Fan or surrounding sheet metal is probably bent from having intimate relationship with mouse. You should be able easily restore things to their normal state with your fingers. No professional services required.