Okay, so we get a lot of UFOpaedia reports about the various items that use dimension chambers for energy (Alien Energy Source, Disruptor Gun, Devastator Cannon, Dimension Missile Launcher, Vortex Mine, Light Disruptor Beam, Medium Disruptor Beam, Heavy Disruptor Beam) and a report on Dimension Gates themselves. I think I've pieced together what's actually going on here.
The Alien Dimension is full of antimatter. Every object we see there - the planet, the buildings, the UFOs, the air - is antimatter, not matter.
This immediately explains why trying to take an ordinary craft through a Dimension Gate results in catastrophe. The air on the other side is made of antimatter; your ship will explode in an enormous fireball on contact with it. The Alien Propulsion System is a device that somehow flips a ship from being made of matter to being made of antimatter (or vice versa) as it goes through the Dimension Gate, allowing safe passage (there's a line about "its ability to transform matter into antimatter" in the description of the Alien Control System).
A Dimension Chamber is simply a miniature Dimension Gate that allows a tiny trickle of antimatter to seep through. The antimatter then annihilates, generating energy. Technically, this would use fuel, but anything will do; a small amount of air would suffice. This also explains why they are so explosive; any antimatter stored in the device would immediately annihilate if containment is lost. Of course, that in itself can be useful for a bomb (see: Vortex Mine, which explicitly says it's warping in antimatter from another dimension).
I think this hypothesis brings together all the various hints we're given quite nicely. It does create one slight plothole, namely why neither side simply flings objects through the Dimension Gates to blow up the city on the other side (a Hawk attempting to fly through a Gate should logically create a fireball that makes nukes look small), but I guess the Dimension Gate Generator might be well shielded and the Micronoids wouldn't gain much from blowing up Mega-Primus.