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The net force is the vector sum of all the forces that act upon an object. That is to say, the net force is the sum of all the forces, taking into account the fact that a force is a vector and two forces of equal magnitude and opposite direction will cancel each other out.
Yes of course the object exerts a reaction force on the rope. That is the whole point. An action/reaction pair is always the same type of force, so the pair force of a tension is always also a tension. They also never act on the same object, The reaction acts on the rope, not the object. The only force acting on the object is the 'action' (pull) from the rope. Thus, as Ken said, the object is not in equilibrium under the action of external forces.