To me the idea of a nation of monsters trying to establish itself as a world power is an interesting concept. Are there resources besides the Eberron Campaign Setting for more information about the nation of Droaam? Any D&D edition is fine; I'm more interested in background than mechanics.
This answer discusses the 3.5e sources; I have limited knowledge of 4e’s take on Eberron (aside from being aware of a series of really, really bad ideas being implemented as ret-cons; I don’t like what 4e did with Eberron at all and recommend, at the very least, turning a very critical eye towards it).
Within 3.5e, there is really very little. Faiths of Eberron mentions nothing about religion in Droaam. Magic of Eberron has nothing about the magic going on there. It’s not one of the five nations, so Five Nations barely mentions it (only a mention as a border that Breland needs to keep watch of). None of the Dragonmarked Houses come from there, so Dragonmarked has nothing. Races of Eberron mentions goblins and orcs, but mostly with respect to their greater populations in Darguun, Eldeen, the Shadow Marches, and the Demon Wastes. Secrets of Sarlona, Secrets of Xen’drik, and Dragons of Eberron (which might as well be Secrets of Argonnessen) are all right out.
There is one book in 3.5e beyond the Eberron Campaign Setting that mentions them in slightly greater detail:
Player’s Guide to Eberron
The 3.5e Player’s Guide to Eberron has a couple (as in literally two) pages on Droaam. It details a little bit about the most common monstrous races there (gnolls, goblins, half-ogres, harpies, hill giants, medusas, minotaurs, ogres, orcs, and trolls), as well as a city, Graywall, which is the “Gateway to Droaam.” The races get a paragraph each, the city gets four. There’s also a sidebar with info that you can get about Droaam from Knowledge checks, and another sidebar entitled “Dungeons of Droaam.”
Overall, there isn’t much, and most of it is extremely general excepting the bits about Graywall.
3.5e Web Articles
During 3.5e, a number of web articles were published on Wizards’ website with information about Eberron, mostly the “Dragonshards” series by Keith Baker. I must admit, I really dislike a lot of things within Dragonshards—like many setting creators, it seems (at least to me) that Baker did best when there were others around him to temper and improve his ideas. But they exist, and a few of them even mention Droaam. So with no further ado:
Medusas of Droaam—some details about the history and culture of the medusa population in Droaam.
Masters of Magic—Mordain the Fleshweaver lives in Droaam, and Sora Kell the Queen of Night, of course, is at least nominally related to the Daughters of Sora Kell. Mordain is one of the primary reasons I think so poorly of the Dragonshards articles, though; much of his entry flies in the face of what has been published about Eberron in actual books.
Forward Sora Kell Petition to Thronehold, Karrnathi Ambassador Says—an in-character news article about Droaam’s attempts to be taken seriously as a nation.