@Irian
Regarding the war and the alliance.
It's implied that the Republic somewhat recently turned into a borderline dictatorship, with atrocities, human rights violations (if such a concept exists in-universe) and everything.
The Empire is trying to be a power to deal with, and projects its influence through this war hiding behind a noble cause.
As for the Grand Duchy, I suppose it's in it for a landgrab. The Black Sea coast, especially Crimea was a pivotal point in Russian-Ottoman relations in our history, it's the same in here I suppose. Thwarting the Reds with stricter laws of the wartime is also a major reason. Maybe some economic benefits. Or just simple warmongering, but there is no evidence for that.
The Empire and the Grand Duchy being monarchies (as well as the other mentioned power, the Kingdom) are likely to rally up against a non-royalist country despite their differences.