No, a few survived.
In ST:FC, Riker said there were "very few governments left" after WW III. He didn't say there were NONE left...
And in TNG's "The Royale", there is a United States flag with 52 stars, which (according to dialogue) came from somewhere between 2033 and 2079. WW III took place in 2053, so therefore the USA was one of the nations that survived the war.
No, it wasn't.
The Vulcans may have helped in the cleanup and rebuilding after WW III, but they had nothing to do with creating the state of United Earth. Humans did that on their own.
What government isn't?
Eddington is full of shit.