He'll deny this at first saying Skyrim hasn't even had a true high king in a long time. He used a thu'rm or dragonshout to weaken the king which he learnt from the Greybeards which is a violation of all the Greybeards beliefs in the way of the voice. After weakening the high king, Ulfric plunged a dagger into the king which he admits if you talk to him about it; People are split over if it was a fair fight or a cold blooded murder, it's likely the latter given the weapon of choice.
Spoiler It's rather clear after the end of the stormcloak campaign that the position of High is all Ulfric was really after from the beginning, he intends to use his stormcloaks to force the Jarls to appoint him high king, at least the Jarls likely to oppose his appointment. If any Jarl refuses to accept Ulfric as High King it's likely only death awaits them. Ulfric might be greedy but he isn't an idiot, he plans to ready himself for full on war with the Aldmeri and Ulfric would have no qualms about attacking them if they were to attack Cyrodiil or Hammerfell
So he claims, it's hard to know if he is really telling the truth but his actions do show that he has no care at all about the other races.
Primarily this is the Dunmer who fled Marrowind because of the issues with the Red Mountain which caused a lot of Dunmer to flee Marrowind. The Dunmer alive when this happened kinda deserved it however on the most part, their children do not tho. As for the Argonians in Windhelm, they're treated as virtual slaves and have done nothing to deserve this, if you aren't a nord they don't care. Ulfric also played a part in the raids/invasion on the Forsworn home city of Markarth, which appears to be a generally unprovoked attack against the primarily Breton inhabitants and Nords of skyrim have a horrible history of attacking the Bretons in usually unprovoked attacks.
Imperials are Humans just like the Nords and are of common descent, Imperials tend to care about all the races which can be a weakness sometimes considering a lot of the provinces would like to see the Imperials dead. The Imperials are over-stretched and the Aldmeri Dominion believes all humans have to die, so the Imperials need to harden up and accept that a real war with Summerset Isle may be needed. Not all Altmer support the Aldmeri Dominion tho, some even stand firmly against it and so a war with the Aldmeri Dominion may not be as bloody as everybody thinks it will be.
yes and no, Skyrim has a high king and pretty much rules itself, it is a part of the empire as are all the provinces so the empire does remain the highest authority. I would suggest you think of it a bit like USA but without democracy, the Emperor is a bit like the president, the high king is state governor and below him are the more regional Jarls. It was the Nords that pretty much set this up to begin with, the only thing that changed is the line of succession for Emperor switched to Imperials instead of Nords (dragon borns are primarily candidates for the position of emperor). The problem the Nords have is the peace treaty, the White-Gold Concordat which is a sign of weakness on the empires part for having even fallen into the position to have to sign that. Hammerfell also are not happy with the Imperial Empire over that either and feel the Empire abandoned them in their moment of need.
Not exactly, they signed the White-Gold Concordat with the Aldmeri Dominion and are pretending it's banned in-line with the White-Gold Concordat which prohibits Talos Worship; however the empire has made no real action in the enforcement for it and so the Thalmor are having to do that manually themselves. Everybody knows they is going to be another war and the Thalmor realize they can't really enforce the ban on Talos with the empire doing effectively nothing, instead they're trying to use it to force a loss of confidence in the Imperial Empire. The harder the Thalmor push the more the Nords react but they react against the Empire instead of against the Thalmor. The stormcloaks unfortunately aren't very bright else they'd realize the deeper strategy at play, however racism is also fairly common among the stormcloaks and it shouldn't be a surprise they're not the brightest of people in the world.