For camping, if you're on the grid, you do not need a battery. The places I like to camp don't usually have grid access, but a state park is a good place for a shakedown trip.
If you have electric brakes you may need an onboard battery for emergency activation.
The interior lights will work on both 12v and 120v, and yes, you can use the camper plug on a washer outlet. You may also be able to find an adapter to go down to the standard wall socket. Best to be prepared, you never know what kind of receptacle you might need, and the adapter could make things easier for you. (My camper lives in my backyard, so if I want shore power I run an extension cord.)
1995 Toyota 4Runner SR5 4WD 5 spd
2002 Eagle Summit 12