Every apartment I've ever lived in has been painted before a new tenant moves in. Usually long-term tenants can get free painting every so many years - you could check with your landlord if you've been there for a while.
Kids are actually great at washing the lower parts of walls! Just don't give them magic eraser - the stuff is non-toxic but if they rub on skin it can be highly abrasive. My 4yo likes to help, and his 7yo brother likes to earn spending money with extra tasks like that. They are great at washing the fronts of the lower kitchen cabinets.
In my experience, landlords use really cheap paint, so start with a gentle mix of dish soap and water and a non-abrasive sponge.