There's one important precondition that needs to be understood before the question of a "galactic empire" can be addressed. So far, nobody but @user11599 has mentioned it (+1 for doing so.)
An empire is a predatory economic structure.
More precisely, one of the best definitions for present purposes comes from John Michael Greer's useful article The Nature of Empire:
An empire is an arrangement among nations, backed and usually imposed by military force, that extracts wealth from a periphery of subject nations and concentrates it in the imperial core. Put more simply, an empire is a wealth pump, a device to enrich one nation at the expense of others. [Emphasis added.]
From this perspective, it's easy to see that the traditions of Science Fictional worldbuilding have more often than not put the cart before the horse, concentrating on the trappings of imperial prosperity rather than the underlying wealth pump that provides the reason for the empire's existence, and that makes it all possible.
True, there are some thoughtful exceptions - Dune's commercial empire, for example, and the various interstellar states of Asimov's Foundation series - but usually we are shown mighty fleets, militaristic cultural tropes, dizzyingly wealthy stellar imperial capital cities, and so on. These interstellar empires are (especially in MilSF) often imagined as the pageantry and politics of some historical terrestrial empire (commonly British or Austro-Hungarian) transposed to the stars. The wealth pump is handwaved, or simply ignored.
However, @HDE 226868, you are doing something that none of your imperial forbears in the field has attempted: you are investigating the feasibility of a lightspeed-bounded interstellar empire. Your list of concerns (communications, troop deployments, longevity & stability of distant civilizations etc) is a good start.
However, you are ignoring the elephant in the room: sublight travel even to a nearby star such as Proxima Centauri is shockingly costly. (See Charlie Stross's The High Frontier, Redux for a brutal but very well-founded description of just how hard and costly sublight interstellar travel would be. Also, scan the comments, and read the shrieking denunciations by fans. Most enlightening. When people are saying to Charles Stross - Charles Stross - "And you call yourself a Science Fiction writer!" you can almost taste the tears through cyberspace. A serious endorsement of the strength of his article: these people, emotionally dependent on the narrative of human colonization of interstellar space, are (understandably) losing their shit... which underscores my present point.
In sublight travel, you don't get to build a reasonable ship and send it through a wormhole, or engage warp drive; the thing has to be big and expensive. Even fuel for the trip is costly. You're building something large and specialized, that will take a lot of fuel for the trip. It might be a relativistic time-dilation vessel, a huge generation ship, or a large and difficult coldsleep ship - and we don't yet even know how to do coldsleep.
And then, no matter which type you build, it's a significantly long voyage, with severe challenges at the destination. (See James P. Hogan's A Voyage from Yesteryear for a rather nice exploration of the social/military disadvantages facing a sub-c expedition upon arrival.)
Which brings us to the core question. What kind of wealth pump can be constructed, sustained, and defended under these conditions?
More importantly, what possible kind of wealth could such a pump possibly transfer, to make such a huge and costly effort worthwhile?
These questions are fascinating and powerful, and very possibly disturbing. Great stuff for a story.
However, it won't be trivial to come up with Worldbuilding.SE-grade answers. :-)