"Alternatively, what if there were four engine rooms? There were 4 lithium circuits that were blown in "Mudd's Women" and the ship in "The Alternative Factor" required 4 dilithium crystals to operate at full power."
According to the above quote from "Mudd's Women" there seem to have only been three and Scotty suggests that, too:
SPOCK: What's wrong?
SCOTT: I don't know, sir. With those three lithium crystals gone
SPOCK: It'll take longer on battery power.
MCCOY: Never trusted this.
"One of the reasons that I chose to have all four crystals in one place (or one room) was because there was only one crystal converter assembly in both "Mudd's Women" and "Elaan of Troyius" suggesting a single point of failure rather than 4 different points that can fail."
There was only one crystal converter assembly in "Mudd's Women"? Indeed, "Elaan of Troyius" suggested one "dilithium crystal converter assembly" (and I'm still wondering how they defused the "bomb" only to find out after that that the entire assembly had been "fused"
"Also, not all Season 2/3 engine rooms had the slide-up crystal holder. I think there was one episode that had some other rectangular device there instead. So you might need to have enough engine rooms for each crystal PLUS one more that doesn't have it. Am I correct to assume that in the engine room per crystal scenario that the Season 1 engine rooms are still also on the ship?"
You're right, it appears the rectangular box was there throughout the second season starting in "The Doomsday Machine"
and in "The Changeling"
and in "Journey to Babel"
and in "The Immunity Syndrome" (lower right side)
and in "The Ultimate Computer"
It appears that the central engine room in the engineering hull doesn't have it or didn't prior to Season Three.
We might, of course, be looking at a practical upgrade: Originally, you had to take the matter-antimatter reactors offline to remove and replace dilithium crytals, maybe through that hatch or drawer below Scotty's communicator:
Maybe the upgrade enabled the removal and replacement of one dilithium circuit at a time while
the reactors are working. Scotty suggested to do that in "The Paradise Syndrome" but Spock felt the replacement procedure to be too time consuming. In "Elaan of Troyius" they were merely able to replace one circuit with the raw dilithium crystals but obviously enough to give them warp power (they used photon torpedos to disable the Klingon Battlecruiser).
I assume the Season One "Engineering Control
Room" to still be on the saucer's port side and wonder if the one from "The Immunity Syndrome" might possibly be the one on the saucer's starboard side (I usually associate the grey housing on the engine room's floor with warp drive functions but the console from the Auxilary Control Room does a great job hiding it from sight...).
After all, the idea was to combine warp and impulse propulsion into one major forward thrust. I'd like to believe this required a manual override (i.e. to add impulse power) and that this was what Scotty had been doing in "The Immunity Syndrome".