Still, a name that's been in power for thousands of years carries a LOT of weight. The new warden has been there for less than a year (if that) and has a certain reputation. Whoever begets an heir with Sansa gains at least a degree of legitimacy, even if grudgingly, as it at least carries on their beloved liege lord's bloodline.
She was always seen as the wronged one within the northern rebellion. Remember, ever since Ned got killed one of the main motivators was "getting the girls back" , followed by the newly claimed independence.
Why would you say that? Stannis picked the niece's letter because EVEN a girl of ten wouldn't give him the time of day. I gather that wasn't the only exchange he's had. (Consider Jon's reaction: "northerners can be a bit like the wildlings", Northerners, not Mormonts) Remember, the red wedding killed other northern lords. As Tyrion said right after the event: "The northerners will never forget". Lets not forget the Umbers who are presumably hiding that other, MALE, key to the north.
As for the Mormonts, remember what the Lord Commander said to Jon when he handed him Longclaw: "It was meant for my son Jorah. He brought dishonour to hour house but he had the grace to leave the sword before he fled from Westeros.". That implies that they thought Ned Starks ruling on Jorah just. It's how you get a former head of a house giving his former liege lord's bastard son a sword that has been in his family for five centuries.