I believe that Ulfric Stormcloak is an agent in the literal sense of following their orders, but in the sense that they forced all nords. into war with the signing of the White Gold Concordiate and the outlaw of worship of Talos. With the war, they can virtually move undetected in Skyrim. If the the Stormcloaks win the conflict, Highrock will be cut off of supplies from the Empire. They can't go through Skyrim do to anomosity towards the Stormcloaks, and the can't go through Hammerfell with supplies because they'd never get through the Desert in time to help Highrock. With Highrock cut off, Thalmoor can take Hammerfell. After the fall of the Redguard, the Thalmoor can move onto Highrock an just over power them with time. They can then retake Skyrim, the Thalmoor can then cut Cyrodil in half by assualting on four fronts: Valenwood, Elsweyr, Hammerfell, & Skyrim. And this all started with the outlaw of Talos worship.
If the Empire and the Stormcloaks stopped the Conflict outright, that would go badly for the Thalmoor. Their whole plan is on the Stormcloaks Winning this Particular war. If the fighting ceased, the empire would have a much better chance of defeatingr the Thalmoor.
The Alik'r desert doesn't stretch that far, it doesn't reach eastern Hammerfell, although I doubt the Redguards would be much friendlier to the Imperials than the Stormcloaks. And High Rock isn't utterly cut off, you still have Naval trade and the EEC, although I don't know what resources High Rock is dependant on the Empire for. And please explain how cutting off High Rock would lead to the downfall of Hammerfell? The Thalmor couldn't defeat Hammerfell 15/20 years ago, they sure as hell couldn't do it now, Hammerfell has a more stable government and better resources (human nations recuperate faster than elven nations). I also doubt the Thalmor would be able to conquer Skyrim, either.
The Dominion lost the majority of their forces from the Great War and the war with Hammerfell, they couldn't afford to launch an assault on Cyrodiil on 6 fronts. In addition, a Stormcloak victory is just as unfavorable for the Aldmeri Dominon as an Imperial victory, in the Civil War.