Sawdust build-up can create heat and friction when you use a hole saw. One solution is to drill a smaller hole inside the area to be cut out that will allow the sawdust to escape. Size the relief hole according to the size of the larger hole. If, for example, you are cutting a 3-in. hole, drill a 1-in. relief hole. Don't drill the hole too close to the center, which you need for the saw's pilot bit.
Wallpapering can involve a steep learning curve and most professionals advise starting in large rooms. Save bathrooms for last, or at least until you’ve got your confidence.