Howland Reed is alive and probably knows the truth because he was a close friend of Ned. With Meera heading back home, I would be surprised if Howland doesn't show up next season.One, does Jon's parentage mean anything to anyone else other than Jon. The info about the annulment won't say "but the kid was stolen and renamed Jon" and no one is going to back up the boy in he wheel chair's dreams. There's really no one alive that could back up Jon's claim to the throne.
They can build a cage on site. Alternatively, use rope.Two, what good will come from capturing a zombie unless it was alive? I didn't see them take a cage or something to put it in and if they bring it back dead, it's just some dead guy. Not to mention, bringing it back alive(ish) seems impossible.