
The Japanese refer to the crane as “the bird of happiness;” the Chinese as “heavenly crane” believing they were symbols of wisdom. The powerful wings of the crane were believed to be able to convey souls up to heaven and to carry people to higher levels of spiritual enlightenment.. Hope this will help a little bit

Yes. Exactly. Paper folded like the birds. The belief is that if you fold a 1000 cranes, you get a wish fulfilled. The subversion here is that he didn't need a 1000 cranes to get a wish. He pretty got everything he wanted in life with just one "Craig." Craig made his wish come true. So essentially Craig is his thousand cranes.
You are my only crane. You are my <cue title page> "a thousand cranes." That was pretty brilliant.