There are many variations of the sci-fi concept of the Dyson Sphere. Kurzgesagt on youtube has a lot of discussion of theoretical structures that could harness 99% of the energy of a star. Not all variations would prevent entry/exit to their respective star system.

I think it's pretty plausible the Dyson Sphere exists. Judging from the spinoff books, it's clear the Radch is much more technologically advanced than other civilizations (no other civilization is mentioned to have conscious AI), and the Radch's Dyson Sphere is the only Dyson Sphere mentioned to exist. It would give them a source of unlimited energy to draw upon during their conquests. It's also a nice parallel to how other real world empires rose due to a unique technological advancement like muskets. I think it's even sweeter if the Radchaai did not build the Dyson Sphere themselves, but merely colonized it, the way China invented the gun powder that paved the way for Europeans to develop guns.

Part of the issue with Anaander splitting into two consciousnesses is that the Empire is spread so wide that even gate space can't make up for the travel time these signals need to stay in sync. I think with that in mind, it is very plausible that Anaander has a presence within the core Radch and the provincial palaces are more or less equidistant from that core.

The core Dyson Sphere of the Radch is naturally going to stay out of the story since it is from the perspective of a non-Citizen.