I definitely think that’s what happened.
The showrunners mention the name Night’s King in one of their short videos.
Is there any evidence it’s Valyrian steel that kills them, rather than the particular sword Jon Snow is wielding (I didn’t even know it was made of Valyrian steel)?
My theory is that the White Walkers are empowered by winter. There are plenty of places it seems to me that don’t experience winter, like where Daenerys has been hanging out. So my guess is that there’s a limit to how far the White Walkers can extend their power.
Jon’s sword is Longclaw. Formerly the sword of House Mormont for five centuries. Made of Valyrian steel.
Was it explained in-show how Jon acquired that sword? Was it given to him at the Wall?
Jon’s sword is called Longclaw, ancestral sword of House Mormont. Lord Commander Jeor Mormont gave it to him for saving him from a wight, since his son Jorah the Explorer brought shame upon the House by selling slaves.
White Walkers do this to blades made from ordinary steel.
Yup, the former Lord Commander, Jeor Mormont, gave it to Jon after Jon saved him from a zombie. They showed it on the “previously on” at the beginning of the episode.
If Jorah wasn’t a slaver it would have gone to him.
On the “previously on…” part before the show started, they made a point to reshow Jon getting his sword, with the dialogue of (basically), “Oh boy, it’s Valyrian steel!”
Presumably there are terms and conditions of the contract, just like insurance here. We only heard her side of the story, and secondhand at that. Perhaps the Gambler has reason to believe that the captain scuttled his own (worn-out and empty) ship for the insurance money, collecting payment for his cargo twice, and is refusing to pay out for that reason?
Just gotta say I love the nickname for Jon Snow - King Crow
Except we don’t know how far south Winter can extend, once it comes! I’m getting the feeling that this is not your grandfather’s Winter…
I was seriously thinking the night king would freeze the sea nearby and the zombies would start heading out…
Or would that just be too cool, and things ending rather suddenly for the rowers…
amazing the WWs just showed up when Snow and the other guys from the watch were visiting. 
I imagined the finale battle at the wall and one of the smaller dragons sweeping low by the wall torching wights, then a horde of them leap off the wall onto the dragon’s back and take it down. A dragon riding Dany tearfully commands Drogon to torch the remains before it can reanimate.
So does that mean Jon Snow is now daddy/uncle/ward/carer for two young orphans?
and why are the Free Folk wearing Operation Desert Storm surplus kit?
The two Wilding kids? Why would Jon become their caretaker when there are thousands of other Wildlings who can look after them, probably including some who already know them?
Give them to Sam. He doesn’t mind rearing wildling children.
Ok…not much more to add. That was a totally bad-ass episode.
Aren’t we saying the same thing? They’re both undead aren’t they?