Why don't we consider both the forces while calculating the magnitude of stress in an elastic body? Consider a wire being stretched from two ends with equal forces. We know that both of these forces collectively participate in elongating the wire; had there been one force the wire would have accelerated in the direction of force. Why can't then the stress be calculated using the two forces (knowing that the vector resultant of the two forces would come out to be zero)?

 A: Yes, we can calculate the stress in the case you have given, and that stress is still given by F/A. 
Why is the stress for the case in which the wire is attached to ceiling at one end and pulled with a force F at the other also F/A? In the ceiling case, the net force is zero just as in your case. So the force exerted by the ceiling on the wire to pull it must be equal to F as the net force is zero. Thus it is the same as pulling both ends with a force F (as in your case). The ceiling case is therefore essentially equal to your case.
Hope it helps! 
A: Tricky question. Basically you would think the total force is 0 on any plane intersecting the cylinder at right angles, hence there would be no pressure, right? Well, the first point (0 net force) is correct, second is not.
Imagine being physically pulled by two equally strong friends in opposite directions. Total force is 0 so you remain standing where you are, but you will feel a stress (pressure) in your body from your pulling friends.
It helps to think of the situation from a different perspective: One force is trying to pull out (extend) the cylinder, and is therefore providing a pressure/stress on the cylinder, of magnitude P=F/A. But the cylinder is not moving, so there must be an opposite force of equal magnitude, which maintains the cohesion of the cylinder. At some level of external force this cohesion is no longer strong enough, so the external force will rip the cylinder apart.
You can think of the cohesive force as a sort of reaction (or supportive, passive) force, and the pulling as an "active" force.
A: There aren't 2 forces. There is only 1 force, a tension = F. You see the convention of how that force is depicted in a free body diagram. If you draw an axial force diagram for the wire it would be a straight line at a magnitude of F. There is a uniform tension at any point in the wire of magnitude F. The stress (force per unit area) is that force divided by the cross sectional area.
