Actually that is how Skyrim politics are, if you see them unfair and unjust then you're against Nordic tradition. The Jarls are all largely independent and rule their holds, they do however swear fealty to Skyrim's High King, the Pact of Chieftains is a Skyrim tradition which actually Ulfric cuts out by having Galmar control the Jarls to follow orders Ulfric gives (Those are the actions of a Tyrant)
Windhelm is in sorry condition, it's actually bankrupt. Their stockpiles are completely empty, you would know this if you did the Legion quest line and spoke to the new Jarl, Free-Winter. Windhelm is falling apart the walls are barely standing and are in disrepair, Windhelm is as poor as Winterhold just it still has walls. Windhelm's diverse mix of culture? The Argonians were banished from the walls and the Snow Quarter was renamed the Grey Quarter as an insult when Ulfric took over. Jarls before Ulfric cared about other races, Ulfric does not.
'There is cause for optimism, though, as Jarl Ulfric is not nearly so tolerant of these substandard beings as his fathers were.' Also that when Non-Nords are attacked in his Hold he doesn't lift a finger, but when Nords are attacked he is the first to sound the horn. The Stormcloak rebellion is being aided by the Thalmor, take away that Thalmor aid and the rebellion would of ended long ago, Windhelm is bankrupt and has no stockpiles of anything left.
The Empire never struck a deal with Ulfric during the Markarth incident, the Forsworn Kingdom were actually making steps to be recognized by the Empire as a legitimate kingdom, the Empire however was busy since it was dealing with the aftermath of the Great War, none of the Legions had even returned to Skyrim. Ulfric was promised free worship by a few Nords including Igmund, once the Thalmor got wind, they ordered his arrest or it was back to war, no one was willing to sacrifice the newly formed treaty.
It is nonsense, no enemies of the Empire would pass through Skyrim unless it was the Nords themselves, which if they left the Empire peacefully and maintained relations then there would be nothing to worry about. I doubt the Nords are going to let the Thalmor pass through their lands towards Cyrodiil. Though the Nords would never mess with Cyrodiil now, Ulfric himself is afraid of pissing off Cyrodiil, because all of the highly trained legions are in Cyrodiil, not that Auxiliary garrison that is within Skyrim.
"We're ready to march on Solitude, but the Emperor's cousin is getting married! If royal blood was spilt, all of Cyrodiil would be up in arms. We can't afford an all out war with the Empire. So we'll bide our time for now."
"We're ready to march on Solitude, but the Emperor's visiting! The goddamned Emperor! And, as much as I'd like to kill the man myself, we can't risk an all out war with the Empire. We'll bide our time for now..."
Messing with the veteran legions isn't even something Ulfric is dumb enough to do.