Regarding the north being free and not the other kingdoms - The North was a free kingdom until the Targaryen rule. So it is only relatively recently it became part of the 7 Kingdoms so it isn't too surprising for it to be asked and granted that they have their independence back.
The issue is only Sansa ever brought it up. Ned didn't care they weren't free, Robb only fought because they killed Ned and stole the throne and would be bad leaders. Jon didn't care but Sansa randomly kept saying "we need independence". In fact what she did was create a situation where Kings Landing has no power, the king of the 6 kingdoms has no ability to enforce any plans he wants, not any power to simply demand taxes from the other kingdoms. The North has been absolutely mashed by years of war with the Lannisters, then the wildlings, then the undead. They'll be one of the weakest kingdoms out there and she gave up the protection of the king of the 7 kingdoms, to go it alone because something the North never had in her lifetime, that she has no idea about, and having never led anything at all and having the diplomatic finesse of Drogon ice skating (oh, you're about to save the entire north... well **** you for making Jon bend the knee, I don't trust you and want independence... throws herself at the floor and starts wailing while pounding the floor with her hands and feet... I WANT INDEPENDENCE).
AS said before the show ended with a toothless king, a couple of WAY stronger kingdoms with no reason to rebuild Kings Landing on their dime when they can walk in and take the whole 7 kingdoms pretty easily.
Sansa traded the protection of a strong king of the seven kingdoms for independence which should lead to their destruction within months... days if the teleportation spell over Westeros is still active.
Also worth noting, Sansa wasn't the head of the Stark household, Jon was, no one of the Starks thought they were 'above' Ned in the pecking order while Ned was being held prisoner and after jon comes Bran, not Sansa. Sansa literally had no claim and no right to do what she did. Bran was the leader and if he had any kind of leadership ability he would have both said shut up you little twit, you'll be the end of us all but also, I'm the head of house Stark and while I might make you the lady of Winterfell and you can run the north in my stead, I am still the ruler of the North.
The fact that he didn't immediately shows he's freaking incompetent and the rule of Bran is destined to be weeks long.