1) Use 120hz on the desktop. Most games allow you to set the hz in game. If not, 24hz isn't a huge loss. That or you can change the hz when you need to. I bet you could make an easy script to do it at the click of a button.
2) The reason 144hz causes stutter with 60 FPS videos, is because videos use their own form of V-sync, and since 60 does not divide evenly into 144hz, some frames will last longer than others. As has been mentioned above, 24hz does divide evenly into 144hz, which will work better than 60hz. 24hz is standard for movies.