Considering Jon turned down Stanis’ offer about being Jon Stark wouldn’t he just make the same choice when he finds out about Robb’s will? Or is the point of the will to bring tension and undercut what should be a happy family reunion with the remaining starks coming to Winterfell post battle of ice?

Jon would probably feel very different about Robb’s will from Stannis’ offer. 

“Jon.” Melisandre was so close he could feel the warmth of her breath. “R’hllor is the only true god. A vow sworn to a tree has no more power than one sworn to your shoes. Open your heart and let the light of the Lord come in. Burn these weirwoods, and accept Winterfell as a gift of the Lord of Light.”

When Jon had been very young, too young to understand what it meant to be a bastard, he used to dream that one day Winterfell might be his. Later, when he was older, he had been ashamed of those dreams. Winterfell would go to Robb and then his sons, or to Bran or Rickon should Robb die childless. And after them came Sansa and Arya. Even to dream otherwise seemed disloyal, as if he were betraying them in his heart, wishing for their deaths. I never wanted this, he thought as he stood before the blue-eyed king and the red woman. I loved Robb, loved all of them … I never wanted any harm to come to any of them, but it did. And now there’s only me. All he had to do was say the word, and he would be Jon Stark, and nevermore a Snow. All he had to do was pledge this king his fealty, and Winterfell was his…

You can’t be the Lord of Winterfell, you’re bastard-born, he heard Robb say again. And the stone kings were growling at him with granite tongues. You do not belong here. This is not your place. When Jon closed his eyes he saw the heart tree, with its pale limbs, red leaves, and solemn face. The weirwood was the heart of Winterfell, Lord Eddard always said…but to save the castle Jon would have to tear that heart up by its ancient roots, and feed it to the red woman’s hungry fire god. I have no right, he thought. Winterfell belongs to the old gods.

Accepting Stannis’ offer would have meant taking Winterfell without the leave of the Starks and betraying their gods. If one takes the position that death releases Jon Snow from his vows, then Robb’s will not only gives that consent but makes it a command, and Jon would be under no obligation to burn the weirwood. 

However, there’s going to be tension regardless, because there’s going to be Sansa and the army of the Vale, Rickon and the Manderlys, and possibly Bran as well if the most recent episode of the show is any indication. 

what should court in Winterfell look like? Winterfell always seems to me so underdeveloped. Where are Catelyn laides, shouldn’t she have brought ladies with her as well as had northern ladies too? Where are all the squires and fosterings, where are the young ladies to be with Arya and Sansa and to hope to catch the eye of the heir? Is this realistic for the capital of the north to seem more like an out of the way small keep rather than the bustling core of the north?

I dunno if underdeveloped is the right word. Seasonal might be the better description – if you look at Robb gathering the banners in AGOT, or Bran’s harvest feasting in ACOK, the North definitely gets together at Winterfell to politick and intrigue and propose marriages. 

So my guess is that Winterfell’s court in winter is massive, as Wintertown turns into a city filling up with nobles and commons alike. And because everyone’s basically trapped indoors and sharing one another’s company all the time, I expect that you get very intense relationships, both positive and negative. I’m thinking it’s very reminiscent of classic Russian literature, but with more broadswords. 

But I think distance works against having a court during the other time of the year. Rather, House Stark seems to have turned to dynastic marriages and tours of the provinces instead. 

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To answer your other questions:

  1. I think Catelyn didn’t bring any ladies with her because of the war. 
  2. Northern ladies I think it’s mainly the distance issue. 
  3. Squires probably are de-emphasized due to the lack of the knightly tradition. 
  4. As for fostering, Ned seems to have taken against it due to his generation’s experience. 

“Because that’s what Winterfell was built to do.” Drat. How did I not realize that. IT’S IN THE DAMN NAME.

Yep. Winterfell is where Winter (i.e, the Others) Fell. Brandon the Builder built it with the help of the Giants and the Children of the Forest, around the same time he built the Wall. My guess is that Winterfell came first, because you don’t need Winterfell if you have the Wall. (although it could be a Plan B)

Consider certain things about Winterfell: it’s got two massive walls, with the inner having more guard towers than the other; the outer wall is lower than the inner wall, and there’s a giant moat between the walls, not outside the first wall. This is really, really weird as castle-building goes. As far as I can tell, there’s only one reason you’d build it that way (and no, I don’t think the outer walls were built later – the Maesters are wrong about that): because you’re preparing to fight an enemy that you know will not break, will absorb all the losses you can throw at it and swarm over the walls, so that you’re already thinking about fighting at the second line. 

Then add on to it that the whole castle is built around a godswood with a big fucking weirwood tree at the middle, and right on top of giant hot springs that have been piped into the walls so that no matter how cold it gets, the people inside will stay warm and alive. 

Then add on to that this: 

hot springs such as the one beneath Winterfell have been shown to be shown to be heated by the furnaces of the world – the same fires that made the Fourteen Flames or the smoking mountain of Dragonstone…the smallfolk of Winterfell…claim that the springs are heated with the breath of a dragon that sleeps beneath the castle,” and that “the dragon Vermax left a clutch of eggs somewhere in the depths of Winterfell’s crypts, where the waters of the hot springs run close to the walls.”

Winterfell is not so much a building as it is a giant engine for fighting the Others.