"It seems to me there is a scene somewhere in TOS where it’s established that the action is taking place on deck six, and then Kirk gets into a turbo-lift and says "sickbay" and then we see the light panel move slowly once, indicating that the lift is going up, before the doors open again?"
Now that you mention it, I seem to remember the same. Was that "Day of the Dove" where they take Johnson to Sickbay after he had been mortally wounded.

Bob