Problem 1 dISTANCE
The fastest way to go from point a -> b while touching a point on the wall would be to go in a 45degree angle. Going any other way would further the distance due to the fact that any wider angle would increase the line distance making the distance of travel longer.
Problem 2 aNT
The fastest way for the ant to cross from one corner to a diagonally opposite corner would be to find the closest thing possible to a straight line by going from point to point. Straight lines are always the fastest.
Problem 3 bILLIARD write-up
You can hit the ball on any wall twice and get it into a pocket. You just can't hit it in a straight line, that ball won't go in. As long as you bounce it off a corner it will go in. It wouldn't matter which position the ball is places on the billiard table. As long as you bounce the ball off a wall it can go in a pocket. The position of the ball doesn't matter. Though there can be multiple combinations for the ball to go in a pocket, any of them can and will work. There are nearly infinite solutions here excluding the one I've mentioned of hitting the ball straight and there being no pockets for it to go in.