1) Do Thalmor personnel, or even other Altmer treat me differently? Do the Thalmor agents still demand to know whether I worship Talos?
Sadly no, they treat you like any other character and will attack you if you reply wrongly.
2) Do I have access to more Thalmor quests than other races?
No there's only a Thalmor agent in Markarth that gives out a small quest, satisfying but that's basically it.
3) (How) can I avoid Thalmor hitsquads?
I try to avoid the main quest as it will set you against the Thalmor. I only got the hit squads after grudgingly doing a certain part the main quest. The College of Winterhold is another one you might want to avoid at a certain point. And of course the Civil War, neither side should win as we'd like to keep them weakened now do we?
I'm just sad that whatever you do in this game seems to eventually put you in direct confrontation with the Thalmor. *sighs*
4) Is the Priest in Whiterun immortal? 'Cause he's got a death sentence for his speeches!
Hehehe *wrubs hands together* he certainly isn't.
5) Does wearing Thalmor robes or Elven Armor have NPCs treat me differently? So far it hasn't, using unhooded robes and Light Elven Armor (equivalent of Leather armor)
Nope

6) Can I complete Diplomatic Immunity without any violence, or being detected?
I believe it is an option but guess what? That option in the quest is bugged. You should be able to talk with General Tullius for another option to deal with the Thalmor as it clearly shows up in your quest log and I heard PC gamers talk about a note hat exists for that but doesn't trigger any option(see uesp for that).