Is an upward moving object experiencing an upward force?
No. A rock, baseball, or golf ball, tossed or hit upward, experience
no upward force after they leave your hand or the club face.
But an upward-accelerating object does.
An object which is not moving is not experiencing any acceleration, other than the acceleration due to gravity, which, along with mass gives it its weight. The upward force (normal force) acting on the object is equal to but opposite to its weight, and all of the forces acting on the objects are in equilibrium so the net force is zero Newtons.
No. Weight is the perceived force due to gravity pulling down on an object. The force of gravity doesn't change due to an object's motion. The fact that an object seems "lighter" at the end of lifting it is because its upward momentum and the law of inertia causing the object to continue moving upward until the force of gravity stops the upward motion.
It depends entirely on the situation. If the object is resting, the upward and downward forces must be equal. On a planet, like Earth, this would make the normal force (upward) and gravitational force (downward) the same quantity. In outer space, it would simply be inertia. If the object is falling, the upward force is less than the downward force. If the object is rising, the upward force is greater than the downward force.