Do vassals have the right to appeal to the king if they believe that their liege lord is treating them unfairly, or is this a breach of etiquette? If he believes their claim is justified, is there any chance he would risk pissing off a powerful noble by punishing or overruling them in favor of a less powerful one?

Well, as I’ve discussed before, one of the basic goals of medieval judicial reform is to extend the king’s authority down through the layers of subinfeudation, so yes. 

Indeed, I would argue that, since the king’s larger mission is to put and keep his more powerful nobles in check through the law, he’d probably be more likely to overrule them than not.