Answer :
1). Yes. An object can be moving if the net force acting on the object is zero.
A non-zero net force is required to CHANGE the object's motion ... speed it up,
slow it down, or change its direction. But no force is required to keep it moving
at a constant speed in a straight line.
2).a). The motion of the box changes, from not moving to moving.
So the forces acting on it must be unbalanced.
2).b). The motion of the box doesn't change. It goes from not moving
to still not moving. So the forces acting on it must be balanced.
2).c). The motion of the box changes. Its speed decreases.
So the forces acting on it must be unbalanced.