Interesting chat - and before wading in too far I can see that the general impression of the overall approach has some momentum on perception.
First, it does appear that on this ship (but also possibly on the TMP ship as well since there is a full airlock and docking port on the main dome) the dome is a larger structure and the Brigde is not the pinnacle of that structure.
The very top would now seem to be a sensor cluster or something.
What appears likely is that the main deck that has the Bridge also houses a transporter room (probably not a bad idea to have one close by) and some other rooms as well that may have a variety of functions - Ready Room, storage, Equipment, etc.
The Spock shot was (per the commentary) again deliberately filmed to be very very fast as part of the overall idea that the writers wanted to in part play with the audience's perception of the timeline and timescale of the film. However, I don't believe even without this explanation that the area Spock originally left from was in the dome. It's clearly not possible.
I also suspect that Sickbay and the room Uhura was in are also not in the dome and are indeed elsewhere in the ship.
As Horatio observes when they are running through the ship stopping and showing the entrance/exit of turbolifts simply stalls the pace and energy of the scene. And just because they are not depicted in them does not mean they were not. The film has (for pacing reasons I believe) the noted 'jump cuts' all over the place to make it look like things are happening faster than they may actually be.
Bear in mind that this is deliberate on the part of the writers.
Personally I find this a wise choice because including the scenes is really not essential and does slacken the pace during what are intended to be energetic sequences.
So from my perspective anyway - these locations are all mostly separate, I don't need a turbolift scene to rationalise that but certainly that may not work for everyone!
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'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist