I interpret "borders" in space as only being relevant to controlled star systems. A system controlled by the Federation has a 5 light year perimeter around it that is "Federation space" all around it. If you claim 10 star systems in proximity to each other such that there is a very small or almost no gap between them, you can say the entire region belongs to you. Of course, if you don't have an enforceable claim on any of the planets between that cluster and your homeworld, you have an island of territory in deep space that one of your rivals might try to cut off with an embargo or a blockade or even trying to claim intersecting territory to force your ships to spend an extra week and a half flying around it. All of which is annoying and dangerous... not unlike the Klingons.My impression of their border regions is that, unlike the Prime Universe, where star systems are within the lines of a border region, the lines of a border region will cross through a star system. An individual in that universe could be living in a solar system where, like some towns on Earth, it is bisected by an imaginary line and you could cross from the Federation to the Klingon side in the same system. I find it rather silly.
"Plasma conduits" is a technological conceit of the TNG era mainly. As far as I can tell, starships in the 23rd century transfer power using solid state superconductors and high-voltage cables and such.Looking at the cutaway, I do not see how the matter-antimatter reactor connects to the pylons. There doesn't seem to be much of a space in the shuttlebay area, where the pylons joined the hull for the plasma conduits.
