I'm not sure I understand this. Most puzzles are number-crunching but very little to do with graphics (maybe one or two), so no usually OpenGL isn't used AFAIK.
Of course, folks may use it to visualise the puzzles but not to solve them.
You definitely could do it all in shaders. People have done crazier things.
You definitely could do it all in shaders. People have done crazier things.