In each of these cases, the action is the force exerted by the first object on the second, and the reaction force is the force exerted on the first object by the second. The important thing to notice is that action and reaction forces act on different objects in the interaction. Two equal and opposite forces acting on the same object do not make an action-reaction pair!
This is why the two opposing forces don't cancel out. If a football is kicked from both sides at the same time, it doesn't go anywhere, because the two feet kicking it exert equal and opposite forces on the same ball and cancel each other out.
However, when you kick a football with just one foot, that foot exerts an action force on the ball and, at the same time, the ball exerts a reaction force on your foot. In this case, the forces don't cancel out, so the ball travels through the air.
In each of these cases, the action is the force exerted by the first object on the second, and the reaction force is the force exerted on the first object by the second. The important thing to notice is that action and reaction forces act on different objects in the interaction. Two equal and opposite forces acting on the same object do not make an action-reaction pair!
This is why the two opposing forces don't cancel out. If a football is kicked from both sides at the same time, it doesn't go anywhere, because the two feet kicking it exert equal and opposite forces on the same ball and cancel each other out.
However, when you kick a football with just one foot, that foot exerts an action force on the ball and, at the same time, the ball exerts a reaction force on your foot. In this case, the forces don't cancel out, so the ball travels through the air.