All of this could have gone away if the Empire made Skyrim less of a province, and more of an ally of the Empire.
Except Skyrim did not want to leave the Empire. In fact, Ulfric was still loyal to the Empire a year or two after the White-Gold Concordat was signed. Even the talk of independence itself wasn't raised officially (Subtle hints really) until well over a decade after the treaty was in effect.
I don't hate Tullius, at the end of the day he doesn't have a lot of power, he's just directing the legion. I guess we'll never know exactly what happened at Helgen, but Tullius could've incarcerated Ulfric in Skyrim, no matter what I don't think you should execute a man without trial, and even if it was an effort to ensure the Thalmor wouldn't free him, why were they going to execute all the other Stormcloaks, and two men in rags? Why wouldn't they execute Ulfric first instead of the "for the love of Talos shut up" guy and the last dragonborn before him when you believe the Thalmor are circling like vultures? You're asking for trouble. And Ulfric doesn't accept Tullius' surrender because of what happened at Helgen, it was payback.
You have to look at how law is done in Tamriel. Since the First Era, every province operates under the phrase "All are guilty, until proven innocent"
Ulfric killed Torygg, and raised a rebellion against the Empire. He is being executed for that. A trial would be fairly pointless, since Ulfric isn't denying he killed the King. It is something he champions in his cause. If Ulfric did deny he killed Torygg, then you may have a trial. He's facing the exact same thing that happened to an Elder Council member, who was executed in the fashion Ulfric nearly had. Summary execution soon after arrest.
As for not executing Ulfric first, they weren't in a real rush. Ulfric was bound and gagged surrounded by an entire garrison, inside a Legion fortified village. No one was expecting a Dragon.