I know you gave us a pretty good description of the exterior of the ship, but I'm having some questions regarding the interior.
According to your description, our 4 Klick long ship rotates in order to produce "gravity" (actually centrifugal force, since we're technically being thrown at the floor, away from the core). Therefore, it seems to me that our ship could be built in a circle within a circle within a circle design (or tube withing a tube), with the more critical components in the center to minimize risk. If we assume that the engines and engineering, the reactor core, and the engines make up the center (despite engineering being deck 1 and the core being deck 7), then that leaves 8 more decks to figure out the order for. Here's my suggestion:
Outer Hall and Armor
Storage/Hanger Bay
Living Quarters
Sickbay/Communications/Armoury/R&D/The Bridge
Engineering
Center: Reactor Core
Engineering
Sickbay/Communications/Armoury/R&D/The Bridge
Living Quarters
Storage/Hanger Bay
Outer Hull and Armor
The reasonings and explanations: Pretend the above is a cross-section of the ship. Past the outer hall we have the Hanger Bay and several levels of storage. The further away you go from the core, the bigger the decks have to get, so it stays a circle. This means we have a crap-load of storage, which can be sealed off with pressure doors in case of exposure to hard vacuum, much like modern day cruise ships. It also provides great protection for the next level down, the living quarters. My only defenses for making the living quarters so close to the outer hull is practicality, and space. Because the living quarters have the few level deep storage space above, it should be fairly well protected. It would also provide plenty of space for the large amount of crew necessary to keep this behemoth functional.
Next down we have Sickbay, Communications, the Armoury, R&D, and the Bridge. I lumped those onto one circle because they really don't need the same amount of space as the above levels. Instead, they should be sectioned off as you move horizontally down the tube. The Bridge should be at the front of the ship, followed by Communications, Sickbay, the Armoury, and R&D. That puts Sickbay in one of the safest positions of the ship: nearly dead center.
Next is Engineering, which should have it's own level as it (and the Core) form the hub of the ship, and monitors everything related to it, power-wise. The engines, fed by the core, are at the rear of this tube.
Elevators appear at regular intervals throughout the tubes, making it fairly simple to move about the ship, despite its rotation.
Now that I've burned out my brain building a star-ship from the ground up, I leave it to you, the reader, to correct, flame, etc.