Screws can often snap or shear when tightening or undoing them in wood, this video shows how to removed the snapped screw and re-plug the holes using dowels to create a quick and permanent repair. Some screws are more prone to shearing or snapping than others, decking screws for example often shear as soon as you attempt to undo them.
These videos might also be of some use-
Hinge drills- https://www.youtube.com/watch?v=dJ_1pbNwA_Q&t
JCB Diamond Tech hole saws- https://www.youtube.com/watch?v=k66K520mWT4&t