Originally Posted by DJ's left nut
Stark's brother Brandon was killed by the mad king; The Starks have always been (to the best of my knowledge), the Lords of Winterfell; never Kings. I've never figured out exactly why Baratheon ended up as the King, but I think it's mostly because he took it. Stark could've been King had he occupied the throne and established legitimacy, but Robert did it and Stark evidently preferred it that way.
The white guy is a former Knight for the mad king that was exiled. He is essentially a protector for Dany that appears to be loyal to her because of his history with her father.
I thought Stark mentioned to the old guy at the table (when they were discussing how much debt the throne was in) that he served under his (Stark's) father, and then served another King (said in kind of a snarky way like he was pissed about it). And then that guy handed him what I take was the "hand of the king".