Greg, with regard to using a drill to rotate the stump of the screw clockwise and screw it right through the part, there is an exotic alternative. You can buy anticlockwise drill bits. If you are going to have to drill down the length of the screw-stump anyway, to apply a stud extractor, it makes sense to use an anticlockwise-rotating drill with an anticlockwise drill bit, because there is a fair chance it will unscrew the stump far enough to enable you to grab the end of it with vice-grips. If it doesn't, you haven't lost anything anyway except the higher cost of the anticlockwise drill bit (which is re-usable of course).
I haven't personally used anticlockwise drill bits, but they exist, and I have seen this trick advocated by experts.
