If all actions have exactly equal and opposite reactions, why doesn't the universe collapse?

You are right - all forces come in equal and opposite pairs.

When, however, you are considering the equilibrium or motion of a body only the forces exerted on it are relevant, not those it exerts on other bodies. Regarding your question why Newton's third law doesn't make the universe collapse, you only have to consider the motion of the Earth around the sun. They each exert equal and opposite forces on the other, and would collapse into each other if the Earth were stationary. By orbiting the sun it both keeps falling into the sun and avoids actually reaching it, just like a satellite orbiting the Earth both falls continuously and avoids crashing into it.

Newton considers the motion of the system under study and an unbalanced force on it. That does not contradict the fact that the system itself will exert a reaction force on whatever is causing the external force...

The third law is often poorly understood (maybe because many teachers poorly understand it themselves...) . If you push your foot backwards on the floor, the floor pushes you forward. Nothing is balanced there: you push on the floor, the floor pushes on you. Balancing of forces happens only when we consider forces on the SAME object.