Can you throw a ball so that it goes 200kmh/h? Probably not. But, if you're being driven in a car that goes as fast as that and then throw a ball (ignore the air resistance -- they don't have that in space), The ball will have a speed of 200km/h plus whatever you give it.
In essence, anything given a speed x will travel relatively (meaning compared to the thrower) at that speed, but absolutely (meaning in comparison with the surrounding space) at speed v+x, where v is a speed of a thrower.
This explanation, of course, works fine with low speeds (200km/h is slow in physics). If you want to know what would happen at warp speeds... well, you'll have to wait for physicists to find that out.
In Star Trek universe, the explanation of how is faster-than-light travel even possible is as follows:
[...] warp drive technology creates an artificial "bubble" of normal space-time that surrounds the spacecraft [...] Consequently, spacecraft at warp velocity can continue to interact with objects in "normal space".
What is not obvious here is what happens with an object that flies out of that "bubble". The same article also mentions a theoretical solution for such flight, which might help us:
A theoretical solution for faster-than-light travel which models the warp drive concept, called the Alcubierre drive, was formulated by physicist Miguel Alcubierre in 1994. Subsequent calculations found that such a model would require negative mass, the existence of which has never been supported by any evidence, and prohibitive amounts of energy. However, it has recently been found that by changing the shape of the warp drive, much less negative mass and energy could be used, though the energy required is still many orders of magnitude greater than anything currently possible by human beings. NASA scientists have begun preliminary research on such technology.
So, I assume that an object flying out of the "bubble" would have negative mass and huge amount of energy, but upon entering the other ship's "bubble", it'd probably again have positive mass and its normal amount of energy. In other words, since both ships are traveling under the approximately same conditions (and through a vacuum), faster-than-light speed would not affect them much.
Also, what is the size of the "bubbles"? Maybe they are big enough for the two "bubbles" to have intersected, so the projectiles have never left both "bubbles"? Are these "bubbles" what we see as a tunnel? I don't know; I'm not that familiar with Star Trek, but if this is the case, then the whole thing is equivalent to ships not moving at all.
By the way, I have another problem with Enterprise "falling" out of a tunnel ("bubble"?) and getting to an abrupt stop. There is no friction in space and the physics of that stop looked very much like it was a car on a road. Also, you'd expect at least some broken bones with a halt that hard (I'd expect the "bubble", if it was not what we saw as a tunnel, to have popped when the ship was incapacitated, or there was no reason for the ship to stop).