Yeah she doesn't lack warmth, but she's too regal all the time because she thinks that's how she's supposed to act.
I think she's just extremely suspicious of anyone she doesn't know (and even those she does), which is perfectly reasonable after all the shit that happened to her. She doesn't know Dany's motives yet nor can she judge her character and she isn't going to take anyone's word for it, not even Jon's. The standoff-ish tone I think was a deliberate choice by Sansa, she seems to want to poke people a bit first to see how they react. Being nice isn't really high on her list of priorities right now.
1) Where did they say that?
2) So can Jorah, Tormund, Beric, the Hound and that random wight they captured. They did so because Drogon let them, and the will of the dragon is ultimately the deciding point. As for the NK, he's technically riding an undead dragon who has no will, so I don't think any conclusions out of the NK riding Vyserion's body are valid other than dragons can be turned.
3) Nice observation. Maybe there's something to it? It could be the NK's version of religion would be my first guess.
Just watched it again, knowing exactly what would happen - and it still got me. Perfect timing, just a tad before you'd expect it to happen.