Whatever the dilithium crystal does, it should be used to explain how warp plasma is made with what would ordinarily just produce energy. As for the locations of the reactors, Scotty does mention there being crystals in either the nacelles or a separate room of some kind in "The Terratin Incident" in TAS.
Plasma is just highly energised gas; stick some spare deuterium in there and vaporise it, add the energy output from the M/AM reaction and bingo! Plasma for use in the warp engines (AKA warp plasma).
As for
Terratin Incident, the location of the crystals being close to the Engine Room is nothing new; whether those crystals have anything to do with the main reactor on the other hand (like with TNG) is less clear in TOS...
The problem with Scotty doing the work in a service crawlway
in a warp nacelle is that he had to place explosives in key areas to separate where he is from the ship. Since we know that the nacelles can be jettisoned then to me he's operating in a pod that isn't normally separable. And since the engineering hull looks suspiciously like a pod...
SCOTT: I've sealed off the aft end of the service crawlway, and I've positioned explosive separator charges to blast me clear of the ship if I rupture the magnetic bottle. I'm so close to the flow now it feels like ants crawling all over my body.
The reason we only know that nacelles can be jettisoned upon command is from a line in
Savage Curtain though. Prior to that Kirk talked about "discarding" the warp drive nacelles and leaving with just the "main section", but that can easily achieved by severing the dorsal (Scotty thought this was an incredibly dangerous procedure BTW).
Also, bear in mind that in TWS Scotty is not in a general access maintenance tube - it's a service crawlway that leads directly to the centre of the M/AM integrator. Also in TWS Scotty placed (or at least attached detonators to) explosives to blast the offending section clear of the ship. I imagine that he and his men did that prior to him entering the tube, because once he's in there setting explosives will not blast anything clear of the ship; it will just create a big explosion inside the cramped compartment he's in

Clearly in TWS the option to blast away a nacelle is not standard procedure as it requires manual arming the jettison explosives. Maybe Kirk decided to keep the remote activation system on the bridge following the events of TWS?
You would think it was to increase efficiency of the reaction but the dialogue specifically says "accept more reactants at a faster rate of injection". The only way to read that is "inject more fuel to react". And by adding more fuel you should be running out of fuel quicker, not extending it.
LAFORGE: ... Leah, I want to find a way to supplement the energy supply to the ship and to the engines. Could we alter the matter-antimatter paths?
LEAH: Theoretically, yes. The system should be able to accept more reactants at a faster rate of injection
I really wish that scene was better written! As it stands Geordi is seeking a way to add more energy to the ship; Leah seems to be saying that a higher power output can be created by pumping more fuel into the reaction chamber. While technically correct in what he asked, it does so by exhausting their fuel faster - on a surface read, anyway.
When saying "more reactants" what she ought to have said was "more reactions", ie getting more bang from the same amount of buck by spreading the M/AM streams out thinner across more dilithium facets.
This is the only way that Geordi's line "we've found a way to extend the matter-antimatter energy supplies" makes any kind of bonkers sense...I think
What we do know is that if all the dilithium circuits blew out the crystal converter assembly would fail-over to the bypass circuits. So let's say we pulled all the dilithium crystals and triggered the bypass circuits in the converter assembly. Seeing that in "Mudd's Women" the excessive energy load fused the bypass circuits we can assume that excess energy from the runaway matter-antimatter reactor would do the same. Once fused, the reactor's energy has no where to go and then boom!

Of course in "Mudd's Women" they had control of the reactor and we can assume it was just simply idled.
I think I was slightly cross-talking over different engineering setups befofe. Is your reactor in the secondary hull one that mixes M/AM and then sends that output THROUGH the dilithium circuits, creating more antimatter AND converting some of it into shipboard electricity AND sending the rest up to the nacelles? If so, how does the converter assembly play into this? Does it convert the output of the M/AM reaction like in my setup? Does the M/AM reaction still occur inside dilithium in your model or can it be contained by magnetic fields like in mine?
Since this is all based on an off-hand comment I made as an alternative option for a scenario I don't believe in, I'm content to let it go. I'm aware I've derailed this thread a lot already.
One thing I do know for sure - we see Scotty remove a crystal for inspection in
Paradise Syndrome so we know it is possible while the engines are running at high speed.
In-universe I would argue that is not the case. Kirk and Spock never once disputed that fuel would be a problem.
...
Well the plug in plot device helps! (But Kirk and Spock are not aware of the device until after they get onboard the ship.)
Maybe not, but Rojan does tell him that they will be modifying the engines. Perhaps all that is required for the higher speeds is a better rate of energy absorption by the warp coils? Here is the whole exchange:
ROJAN: Your ship, Captain Kirk. It will serve us well in the long voyage that is to come.
KIRK: Voyage? Where?
ROJAN: To your neighbouring galaxy, which you call Andromeda.
KIRK: Andromeda? Why?
ROJAN: It is our home.
SPOCK: What brings you here?
As per usual, Spock's interest in the situation leapfrogs over the Kirk's mundane concerns (for an act that can't happen) onto a subject which deals with deals with a fact at hand - why are the aliens here? Anyway, Rojan answers Spock's question and the ship is captured:
KIRK: What's the point of capturing my ship? Even at maximum warp, the Enterprise couldn't get to Andromeda galaxy for thousands of years.
ROJAN: Captain, we will modify its engines, in order to produce velocities far beyond the reach of your science. The journey between galaxies will take less than three hundred of your years.
SPOCK: Fascinating. Intergalactic travel requiring only three hundred years. That is a leap far beyond anything man has yet accomplished.
Kirk is concerned with the time constraints, perhaps because he doesn't want to die in intergalactic space. Spock (again true to form) acknowledges the impressive ability to cross the galactic void without reference to whether time or fuel was the previous most important barrier to such a feat.
In DOTD, Reactor number three is in the "engineering section" which coincidentally is also what Kang captured in the engineering hull...
As mentioned upthread, the majority of evidence in DOTD points to the Engine Room in that episode being in the saucer...precisely where I would expect Reactor #3 (the Impulse reactor) to be!
Presumably there is a way to convert matter into antimatter for use in the engine.
As for where the matter comes from, it would have to come in from outside through the bussard collectors, or else the matter fuel would run out too.
Even though thin, they are surrounded by matter. It's called interstellar gas. As long as you can extract it from space, then you can run forever barring equipment breakdown or crew starvation.
Regarding what the self-refueling starship model would look like and how it would work - it would have to be able to generate an immense amount of energy for a long time yet only require very little volume for matter and antimatter fuel to generate that energy. How it does it of course is up for discussion
The gas density in between galaxies isn't just thin, it's REALLY thin! OK, more accurately the amount of interstellar material within our galaxy (
around 1 atom per cubic centimetre) is ONE MILLION TIMES denser than that in the intergalactic void (
around 1 atom per cubic metre). So, if you are planning on scooping up the fuel as you go then your system needs to be a
million times more efficient than usual. Conversely, Kirk's objection about the journey to Andromeda (it will take thousands of years) is merely
tens of times less efficient that Rojan's engine modifications make it so.
It seems completely bizarre but fuel does not appear to be a concern for either party on the intergalactic journey. Whether this is because the engines are made millions of times more efficient in their fuel collection (but only tens of times faster) or simply because intergalactic travel is easier on the warp engines to begin with (we don't have too many examples to compare it to) I really have no idea.