That's an interesting observation. My main point was that the ball's weight is proportional to the ball's mass. But you are also right in saying that the ball's weight is proportional to the strength of gravity. As we'll see later on in the semester, going high above the earth will decrease the strength of gravity and therefore reduce the ball's weight at least slightly. Those two forces are essentially the same thing.
Gravity is the general concept while weight is the specific force exerted by gravity.
Gravity can do both positive and negative amounts of work on any object. Whenever the object moves in the direction of the gravitational force on it, gravity does work on the object. Whenever the object moves opposite the gravitational force on it, gravity does negative work on the object. Thus, when you throw a ball upward, gravity is pulling the ball in the direction opposite its motion and gravity does negative work on the ball. But when you throw the ball downward, gravity is pulling the ball in the direction that it's moving and gravity does positive work on the ball.