Do the Cullen kids ever get punished or spanked?

No, Carlisle and Esme may seem like ''parents'', but Carlisle and Esme know that Edward and all the other Cullens are equal to them and they are to be treated like adults. They all are over 50 years old.

Just because Carlisle created Edward, it doesn't mean he can control him or punish him. THey are NOT playing parent.