Where did the Orcs stand in the Great War? I know they're Elves, but they're so far detached from the 'pure Elves' as I like to call them; the Altmer and Bosmer (The Dunmer seem like a far-off sub-race, too)

I don't know how the Dominion treats them, but I know that Orsinium was often ransacked by Nords, so I don't know if they like Mankind all that much.

I would have thought that the Elves may have used the Orcs during the war, but I dunno.

When the Redguards and Bretons sacked Nova Orsinium early in the Era, the Empire bought the refugees and survivors enough time to head to the safety of Skyrim/ Northern Cyrodiil. And Skyrim mentioned that they surved as the Legion's shock troopers. (Though there weren't any Orcish Legionaires that I know of.) So chances are, the Orcs fought for the Empire during the war.

The Altmer hate the Orcs and Dunmer even more than they hate men.

Really, why?

The Dunmer's ancestors (Chimers) Left Alinor and abandoned their Aedric worship for Deadric. The Altmer (Atleast the Thalmor) hold a grudge since then. And the Orsimer were cursed Orcs whose god they worshiped was a former Aedra turned Deadric prince when Boethia turned him.

We served the Empire; the Altmer don't like us, and we don't like them.
Plus, from a tactical point of view, the new Orsinium is in Imperial territory....so....yeah.

I imagine that they fought alongside the Imperial forces in the Great War, simply out of respect, traditionalism, and a hatred for Altmer that make the Nords seem kind.

Although Orsinium tends to stay neutral in most conflicts, they were undoubtedly drug into the conflict at some point and most likely fought for self-preservation more than any other factor. Orsinium is a small nation and had prospered to some extent during the Empire's height. Facing the threat of being targeted by Aldmeri expansion, they likely chose to honor their committment to the other remaining provinces and contain the threat of the Elder Scroll's version of communism: snooty High Elves.