In your unknown locations I'll add: main security/brig
Incident" suggested that the brig or some other holiding facility was located on Deck 2. On the other hand, that turbolift ride from the bridge (traditionally Deck 1) and Deck 2 was awfully lenthy.
The Romulan Commander offered to take her rightful place in the brig. Kirk, on the other hand, had Spock escort her to her "quarters", which tells me that they put her up in the VIP quarters; the two poor shlubs, one of whom was still in his Romulan skivvies when the Enterprise went warping thataway from the Romulan ships, they
were in the brig.
I take Kirk's gesture as 1) a token of respect from one commander to another, and 2) a good way to keep the good commander away from a couple of rather cheesed off troops, who might just convince her to join them in an illadvised attempt to escape and seize the ship.
As for the length of time it took for the turbolift to drop one measley deck, maybe Spock was taking the scenic route, while the aforementioned quarters were being prepared. OR,
once she came on board, certain protocols were put in place, whereas it was required to give specially coded, and therefore misleading, commands to any turbolift that any of the aforementioned Romulans happened to be in, thereby at least frustrating any attempts by the commander, or those two other Romulans for that matter, of figuring out the internal arrangement of the ship. "Deck Two" in this case could actually be instructing the turbolift to head to Deck 7, by way of the secondary hull, pausing briefly by the impulse engines.