My thoughts:

High Elves: sounds about right

Breton: I like the knightly inspiration. They do have plenty of skilled mages who are probably more willing to cooperate than, say, the Dunmer Telvanni.

Dunmer: It would be hard to use the Dunmer in a war game without acknowledging the impact of Great House culture. I would expect Houses Redoran and Indoril to form the core with guar cavalry and infantry, House Dres as irregulars (being slavers), House Telvanni to have the most powerful battlemages when and if they bother to show up, and Hlaalu providing whatever troops they want to provide. All of the Great Houses have a private army and retainers to one degree or another.

As for the tribesmen...you really wouldn't see the Ashlander tribes fighting alongside the Great Houses after the death of Nerevar Indoril and the end of the kingdom of Resdayn. Something about the Ashlanders thinking that the Tribunal murdered Nerevar. There isn't a unifying figure who could command both tribe and house until the Nerevarine fulfills the prophecies in the 3rd Era to become both Nerevarine of the 4 tribes and Hortator of 3 houses. Since the Nerevarine promptly takes her/himself off to Akavir, we don't see the Nerevarine actually lead armies.

Imperial: the Imperials are pretty much the Romans in terms of the legions. I think you'd have to emphasize that in a strategy game. The Romans used plenty of auxiliary units to good effect, but the core of their strength was the formations of the legion.

Orc/nords: fine

Redguards: Redguards are the best swordsmen in Tamriel. Sword singing originated on Yokuda and was carried over to Tamriel. The Way of the Sword is pretty well ingrained in Redguard culture. Mounted archers makes sense with some persian influence, but I do think that the swordsmanship needs to be highlighted.

Argonians: don't forget that the Saxheel smashed the Dunmer in the wake of Red Year. Shadowscales would make excellent elite solo units.

You seem to struggle the most with Bosmer and Khajiit. This is probably because both forces usually engage in more guerilla style warfare.

For Bosmer, the Bosmer usually engage in raiding because of the requirements of the Green Pact and the Meat Mandate. According to the Meat Mandate, they have to eat the meat of the slain within 3 days. So, they usually starve themselves before a raid so they can gorge, but they also raid because that way they don't end up with too much meat from larger battles. I'm not sure how you could make use of the Bosmer knowledge of their own terrain, lightning quick raids, and dealing with cannibalistic enemies in a strategy game, but that's the fun and charm of the Bosmer. In open battle, sure, all they are is archers. But I suspect the Bosmer are much closer to the tactics of the German tribes in the Teutoburg Forest than they are to more typical armies.

For the Khajiit, I suggest you look at the lore for the different types of Khajiit, since the form of a khajiit is dependent on the moon phases. Furthermore, the khajiit have a substantial martial arts tradition. Senche-raht, the "battle cats" you mentioned, are actually one such type of khajiit. There are also types of khajiit that are naturally skilled in magic or skilled warriors. The kahjiit aren't all about stealth.

Sound like an interesting though exercise. Good luck!