One wonders what 20 years he's referring to...
If the movie takes place in 2285 (several years after the 2283 date on the ale bottle, to make sense of Kirk's concern regarding the date), then a refit starting in 2270 would indeed make the "almost completely new" ship two decades old, with just a smidgen of understandable exaggeration. Not a big problem there, considering Morrow has every reason to make things sound more extreme than they really are. It's a reverse sales pitch: "You are flying a piece of junk, Jim, and the facts on this are clear! Sort of. Kinda. Anyway. She's old!"
Khan in turn was clearly not marooned in 2270, but a 2268 marooning (during the third year of the five-year mission judging by stardates) would make the deed seventeen years old. "Fifteen years ago" is a valid expression for that.
When the ship is refit for TMP, the reactor equipment is much different and more compact and the entire drive system is revolutionarily different.
Perhaps, perhaps not. The same facilities could easily have been present in the TOS ship already (but they would be "protective suits must be worn at all times" type areas, only accessed for major repairs or shakedowns, whilst Scotty normally worked in the shirtsleeves control rooms).
But there's plenty of precedent for a propulsive refit creating extra room that is thereafter dedicated to completely different purposes, in the form of WWI battleships converted from coal to oil for WWII.
Regarding the TOS bay signage, we now have proof positive that there indeed are multiple shuttlecraft elevators at the forward end!
I mean, the sign clearly refers to something else than the small door that no doubt eventually leads to turbolifts, because elevators =/= turbolifts.
(These elevators just happen to be hidden behind airtight rolling doors, just as in ST5, because otherwise you wouldn't need the special signage. No pressurization problems there. And elevators with doors are much better for pressurization than turntables without.)
As for seeing "WARNING FIRE" written on a wall, I'd definitely turn my shuttle around, turntable or not, and fly out ASAP! Regardless of whether the sign was referring to a spare warp coil pile fire that has been out of control since stardate 0231.1 and still proudly going, or whether this is the location where warning fire will be fired from when you least expect it.