Question

Newton’s Third Law says that forces occur in pairs and the members of any such pair are equal in magnitude and opposite in direction. That means they must add to zero. This appears to show that ∑F must always = 0 and that, by Newton’s Second Law, a must always = 0. Yet we know that nonzero accelerations occur all the time. Explain the resolution to this dilemma.

Answer 1

Yes, there are action/reaction interactions. And it seems to not follow Newton's Second Law. But it is important to remember that the action/reaction pair of forces act on the two objects separately. If you consider one of the objects as a system, it experiences a force by the other object. Newton's Second Law is only valid for forces acting ON an object, not by an object.
If you were to consider both objects as an entire system, then yes, the sum of the forces will be 0. But that would be a useless exercise if you're trying to examine one object.
I think this explanation is clear.