I like where you are going with this, it appears as if it would play quick and bloody. If you want someone to edit the text for you I would happily volunteer.
If its ok I have some questions and comments.
-You did not define what you mean by "when you have more than one ship you have to roll to activate it".
-On the firing table you have 2 no hit results, 3 hit results, and a critical hit result but they are distributed across the die. I would suggest making a result of 1 or 2 a no hit, 3-5 a hit, and 6 a critical hit. This will make it easier to remember without having to consult the firing table and leaves the possibility of modifiers being added to the firing.
-You mention that torpedoes need an acquire roll but you don't define the parameters.
-You make an odd distinction between Disruptors and Phasers. Phasers are a directed "Nadion" particle beam according to the canon. This would mean they impart energy upon their target through kinetic and radiation effects. Disruptors are a directed particle beam as well but use an unknown particle that breaks down the molecular bonds in the target; this probably makes it a Muon particle beam. Both have been shown to have similar effects on both shields and hull so I'm not sure why you chose to have the disruptors only affect hull.
-An effect that I would recommend you emulate is the apparent bleed-through of damage when torpedoes hit a shielded ship (watch the battle scene at the end of Star Trek VI to see what I mean). I would suggest that the unique effect of a torpedo could be that if a critical (6) is rolled the additional hit impacts the hull regardless of remaining shields.
-Did you specify how it is determined which square is hit when the shields are down? If the choice is up to the player than they will naturally choose the most advantageous choice.
-You mention a damage control roll but don't specify its parameters.
-It might be simpler to have the ships have a given shield regeneration value which determines the number of shields that can be recovered each turn rather than a roll. It achieves the same effect without the need for another roll. Additionally, this allows you to have slight variation in the ships. Just because two ships have the same overall shield strength doesn't mean that they can regenerate them at the same rate.
-I noticed that you placed both phasers and disruptors on both the Starfleet and KDF ships. As per canon Starfleet uses phasers banks/arrays, while the KDF uses disruptors. Referencing one of my earlier points I would suggest that you unify the mechanics for the two types of weapons (simply call them Banks/Arrays/Emitters/Energy Weapons/whatever) so that you don't have to worry about it.
-I assume that you intend to work up more designs for the two sides and possibly other factions?
-You have cloaking squares on the Bird of Prey but do not define their mechanics.