Brynden the Bloodraven is very likely to be the Three-eyed crow, and so could not also be Coldhands.
The Night's King was said to be deposed, his name "wiped from the memory of man," which sounds like he was thoroughly destroyed and likely to have been burned if it was known he was consorting with the Others.
Leaf, the child of the forest Bran meets at the cave of the Last Greenseer, says of Coldhands: "They killed him long ago." Will, Waymar Royce and Benjen Stark died/went missing only two years prior, a small amount of time for someone over 200 years old. In an original manuscript, Martin declared that Coldhands was not Benjen Stark.