So as people within the walls of Winterfell are killed in the battle, can the Night King turn the corpses into wights, even at a distance? Because that means an attack from within. And I think the only way to guard against it is to immediately burn the dead.
If it’s true that wights that have been killed by dragonglass can’t be raised again, it would probably suffice to just stab the bodies with it. I’d guess a glass blade would probably leave tiny shards from the edge inside the wound, so that should work the same way.
Or. like Ed said, “The last one burns us all.”
It’s definitely non-contact we see that after the battle of Hardhome.
In season 1 there is also a wight attack in Castle Black. He had blue eyes when they found the body so it’s unclear whether the raising happened before or after he passed through the Wall or what the exact mechanics are.
Fairly close proximity, though. It definitely makes a difference if the range is 100 yards or 1,000 yards.
I am disappointed we didn’t have a scene last week of Jaime asking Gendry to make him a dragonglass hand. I think Jaime running around poking wights in the eye, Three Stooges-style, would have been awesome.
Has it been established that the Night King knows about Kings Landing? He was trapped north of the wall at least hundreds of years before so it may be that he knows nothing about the geography of Westeros.
The extent to which the Night King and his minions “think” or “plan,” and if they do the extent the manner in which it makes any sense to a human, is never really explained. They aren’t an opposing army, exactly, they’re just a force of nature.
I mean, if you want to think strategically, and assume their goel is the extermination of humankind, the Night King should not have an army at all. He should never attack Winterfell, or any other formation of troops, because he’s not fighting a “war” in the sense we think about it. The undead are more of a plague than an army. He should logically have his Walkers and ghouls scatter everywhere and just try to create more of themselves by ambushing unsuspecting peasants, and eventually everyone in Westeros will be undead and if the people inside Winterfell are still there they’ll all starve to death, or be turned when one or two undead sneak in. Tactics, strategy, logistics, knowing the map, none of that should matter when you’re a disease.
The corpses don’t have blue eyes in the TV show when they’re brought in. The fact that the bodies have passed through the Wall also suggests they are not wights yet.
It’s not clear exactly what was going on there, but it suggests the corpses were reanimated from a distance on the south side of the Wall.
Or it suggests the rules aren’t always consistent. I suspect they are a bit malleable to fit the plot needs.
It could be that there is nothing stopping wights from crossing the wall (except the wall itself), and the magic only prevented WWs from coming through. Until Bran and an undead dragon paved the way for them.
Quiet, you!
I would venture a guess that living military is a better fighting force than undead military. A living person should be able to easily do better than a 1:1 trade. A lot better I would think.
The undead win by pressure of numbers. They swarm you, overwhelm you. Think of most zombie stories. Five people mow down a few hundred but get creamed by the thousand more behind those.
I expect the living to be better here too.
White Walkers are another matter. They can fight well but are comparatively few.
A scene I hope we get is an enraged dragon blowing a furnace blast of fire directly into the Night King’s face, but the flames part around him like a stream flowing around a rock.
Then, from behind, a second dragon eats him.
The Night King was a man before he was turned. Now this was thousands of years ago but he would probably know the basic geography of the country. It’s my opinion, though, that he can “see” things like Bran does so he would be aware of the largest city.
Plus, the wights and the Walkers who made them have some sort of connection, so taking that one wight to King’s Landing may have shown the Night King everything he needed to know, including the faces of almost every major player.
He could strap a couple dragonglass knives to his nub. 
Though King’s Landing was only formed like 200 years ago. It may be more likely he’d be headed for Oldtown (or Casterly Rock) if he wants to go around Winterfell.
He can have a good, old fashioned book burning if he goes to Oldtown. I do think the NK is going to make an end run but I don’t think he’s taking any wights with him. He doesn’t need them, he has a coldfire breathing dragon.
Could be that NK thinks he has Westeros wrapped up (and thats not the worst assumption to make) and he’s off making plans to freeze the ocean to Essos or something cray.