Yes. If you buy a ship from the C-Store you can use it on any other characters you have as long as they meet the requirements for the ship (rank, faction, etc.).
You can upgrade the pistol as a standalone item for I think 50 lobi. I'd assume it increases the damage and it MIGHT add another mod to the weapon since in addition to the increased mk level it's going from a rare item to a very rare item..
Prior to Season 5 I think it was, which ever one gave us the redone mission UI, you could use it to access the replay function I believe. Now as far as I know they aren't used for anything outside of foundry stuff maybe.
I admit I might be wrong, but from the way he worded things it sounded like the item was mentioned on every ship which the cannons are and as far as I know the Quantum Field Focus Phaser is not referred to as "heavy" anywhere in game and should only be mentioned as part of the recon Vesta. Now I own the bundle but if I had…
I'm going to guess the heavy phaser thing you read about was the aux dual heavy cannons which come with every version of the ship. All they are are dual heavy phaser cannons that run off Aux power instead of weapon power. They are not limited to one per ship like some other special weapons.