1. Yes you can plug a refrigerator and a washing machine into the same circuit but this is not recommended because you don't want to trip the breaker and lose power to the fridge.
2. The laundry area still needs a 20 amp circuit run non-stop from the panel to a receptacle meant for the washing machine and connected to nothing else outside the laundry area.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.