Do grazing sheep kill the grass?

No, they simply grow back. Not immediately, but they will. That's why farmer are there to take care of their food supply. But it is possible if there's like 100's of sheep and only a small patch of grass.

Yes, sheep will strip a pasture bare if not rotated to another pasture from time to time.They can eat the grass down to a point where the grass needs to be resown.