Nope, they aren't going to be leveled up when the game is turned off. Otherwise, you could reach a high level very quickly.
IIRC each step you make in the game gives them EXP in the day care.
A good idea is to go up and down through the Cycling Road since all u need to do is press up and release for a second, then repeat...
Day Care should be only used for Breeding purposes because it doesn't EV Train the pokemon.
1 Step = 1 EXP. Point
A good idea is to go up and down through the Cycling Road since all u need to do is press up and release for a second, then repeat...