The first photos are broken bolt/screw extractors. You drill a shallow hole in the broken fastener, and turn the tapered teeth into the hole counter-clockwise. The teeth grip the inside of the hole and, with luck, the fastener unscrews.
I used these quite a bit on aircraft Phillips-head screws. There was one size that would work if you drilled out the stripped cross just a little beyond the bottom of the cross recess. If you drilled deeper, the head would often shear off, but at the right depth, it worked almost every time!
Snap-On used to sell these, but they were sourced elsewhere.