I believe the intermix actually takes place below Main Engineering, near the bottom of the ship. The Shaft we see is just a large EPS conduit.
No -- the concept of an EPS conduit wasn't created until eight years later, so that's an anachronistic interpretation. The big glowy shaft was called the "intermix chamber" according to the reference materials at the time. The idea was that the swirling lights we saw represented the actual matter-antimatter reaction taking place throughout the length of the shaft, something that Rick Sternbach referred to as a "swirl chamber" in his notes to me. Which really makes more sense in a way than the "pulse chamber" design of TNG, because in any given collision between particle and antiparticle streams, only a certain percentage of the particles/antiparticles are going to come into contact, so there will be a lot of unused reactant mass left over. An ongoing intermix seems like it would be more efficient at using up the fuel. Essentially the intermix chamber combines the functions of reaction chamber and power transfer conduit, because the reaction takes place along the entire conduit. (
Voyager also used a swirl-chamber design.)
The David Kimble cutaway poster that came out as a TMP tie-in depicted the bottom of the shaft as containing the antimatter pods, although it was unclear on where the matter supply was. Since the top of the shaft connected to/powered the impulse engine, it stands to reason that both deuterium and antideuterium were injected at the bottom of the shaft.
As for powering the ship, it's not canon, but some "plans" suggest a deck or two below, energy is tapped from the intermix chamber to supply internal main power.
That's what I'd assume, but the problem is that the matte shot in the movie where Kirk looks down the shaft from the upper level of engineering makes it clear that there's nothing connecting to the shaft except for the horizontal conduit on the main level that goes back toward the warp nacelles. And the frameworks supporting that horizontal shaft are not substantial enough to make it seem like they contain power taps, though that strikes me as the easiest retcon.
I've never understood the Reactor room. But, if you'll recall from a few TOS episodes, the crystals appear to be separated from the main reactor. Specifically, "Elaan of Troyius".
Not really. If you look at Doug Drexler's
Constitution-class cutaway based on the artwork that appeared onscreen in ENT: "In a Mirror, Darkly Part II" (
archived here), it shows that the warp reactor ran under the floor of the engine room, with the triangular tubes behind the grille in back being the conduits heading up into the nacelle pylons. This means the unit that the dilithium crystals were raised from was actually the tip of the iceberg, essentially a maintenance port at the very top of the reactor, with the actual reaction chamber a couple of decks directly below it.