Answer
Yes. Yes.
Work Step by Step
a. When you push upward on the ball, Newton's third law means the ball pushes downward on you. You are not accelerating, and the net force on you is zero. The normal force from the floor increases as the ball is thrown, balancing the increased downward force on you.
When the ball leaves your hand, the normal force equals mg.
b. When you catch the ball, again you exert an upward force on the ball. By the same argument, the normal force on your feet, exerted by the floor, increases as the ball is caught.