The rope starts accelerating in the direction of the opposing team. It is the vector sum of the force of gravity and the forces exerted by the two teams. The opposing team, since the rope starts accelerating in their direction. No, there is no net force. However, there is tension in the rope.
Tension results from different forces acting on different parts of the body.
Tension can break things. A pure force, i.e. the same force acting on all parts of the body, cannot break things. If instead of pulling on the rope the two teams push on a heavy rock, but the rock does not move, then again the net force on the rock is zero. However, now the rock is under compression . The tension T is a scalar.