After Skyrim gains it's Independence away from the empire does the moot still pick a new High King or High Queen or is it up to the people to choose for themselves?

If the stormcloaks win and skyrim is free from the empire, i believe Ulfric proclaims himself high king of Skyrim.....but its been awhile since I forced myself to go down that path.

Ulfric actually awaits the moot to choose the high king. In dialogue between him and his right had guy they basically say its a formality as he will be high king.

The moot. Of course, Ulfric if the obvious choice - but when he dies, the new High King will be picked by the moot, as tradition dictates.

ulfric the dictator,/tyran has already proclaimed himself high king and the moot is just a piece of formality since all the jarls will be loyal supporters

If when you play and go the stormcloak route then I would assume that the moot is more or less a formality. Ulfric's victory at that point would all but seal the deal entirely, but again thats dependent on player choices. Though it would be kind of funny if you completed the civil war for the stormcloaks and chose someone else

Skald the Elder gets my vote.

I think that only applies if he dies without an heir.

I think it's still called just in case someone else decides they have a claim to the throne, and they can argue their points. (A moot was called upon Torygg's father's death after all. But at that time no one felt it was necessary to challenge torygg's claim to the throne.)

From UESP:

http://www.uesp.net/wiki/Lore:Moot


Emphasis mine. Altho it doesn't say definitively that the Pact is still in effect, I've seen nothing to indicate that it isn't. And it's entirely possible that a Moot is called at some point after any king/queen dies, whether they are deciding on a successor or not. It would almost be expected, one would think that all the Jarls would need to swear personal fealty to the new ruler at some point. Even if that were not the case there would no doubt be a lot of "business" to discuss and everyone would want to size up the new guy/gal.

OTOH that info is apparently taken from something written in the 2nd Era, soo....

Even though there's very little information on the Moot in-game or otherwise, from the scraps and opinions you can gather in Skyrim, it seems to be that the Moot has simply become a tradition each time the High King/Queen dies. Also I guess the Jarls can swear fealty and such during that Moot, but it would be nice to get information about it aside from "it's tradition" that isn't in the form of an in-game book from Redguard. ._.

I vote help Ulfric and than Shout him down, making my Dragonborn the High king in the same way that Ulfric was high king. I am hero in every hold and so the Jarls will likely swear fealty making my Dragonborn the high king.

Maybe its just me, but the last thing I want is to be high king. If by some weird twist of the universe Ulfric survived my stabbing him in the face then I would say he sort of earned the right to be High King. But more to the point as many have pointed out, I would wager that the moot would sort of elect him though it would likely be little more than a formality.