The eggs can seem to take a bit of time, but just run around and train your other pokemon for a little. The eggs actually hatch kind of quick. As far as cut, Purrloin can learn that move, and you can use Cut after defeating the first gym
A pokemon doesn't need the move anymore it seems, when you select it in the party like you would for summery or switch the HM moves it could learn will just be there without needing to teach it to them.
