The Dominion has free reign to travel through any other province it wants other than Hammerfall, there wouldn't really be any logistical chore of getting into Skyrim by land, and having just spent a great deal of their forces chasing out the Empire, it's unlikely they'd be able to repel the Dominion. Keep in mind what little Thalmor presence we see in Skyrim is just that, very little. Numerous NPCs on the Imperial side of things and a few on the Stormcloak side of things fully admit the Thalmor isn't there en masse because the Empire is "doing their job" for them, which previously involved turning a blind eye whenever they feasibly could up until the Civil War.
Oops, sorry. Black Marsh is a part of "Free Tamriel" as well, but they're not overly fond of anyone so I doubt they'd be much help either.

Throw your name in the hat, etc. Either way the end-game of the specific quest would be your character becoming High King. I'm almost always a Nord in my playthroughs.. it could be a race-exclusive quest and I'm sure something could easily be written that ties the Dragonborn to a noble blood line in some way. EIther way, probably a task for a creative modder, anyway. It just feels kind of wasted to have the Jagged Crown and not be able to do anything with it aside from hand it to one of two guys or keep it for yourself (where it still means nothing).
And other minerals. This becomes a BFD if you do the "Season Unending" quest and there is a fight over who gets Markarth. Although I think Skyrim's main "export" is warriors. As for their food imports, if indeed Cyrodiil refuses to trade with an independent Skyrim (I don't see how they can) or it otherwise goes pear shaped, Skyrim will have to learn to get by on their native resources. Medieval Scandinavia did import, but mostly luxury goods; they fed themselves very well on local seafood, grains and milk.
