Do action and reaction pairs cancel one another?
No, they do not cancel because action and reaction forces act on different objects.
Consider a burro pulling a cart. The burro pulls on the cart with a force F. That is the action part of the pair. The cart pulls on the burro with a force -F. How does the burro get the cart to move?
Consider forces on the cart. Gravity's attraction gives it a weight W, downward. The ground provides a reaction force -W, which is upward. So the cart has 2 forces on it in the vertical axis, gravity and the ground's upward push. They cancel.
Now look at forces acting on the cart in the horizontal direction. The burro is pulling. And ummm ... The burro's force seems to be the only horizontal force acting on the cart. Therefore the cart starts moving.
I hope this helps,