Have you ever seen a bird put its head under its wing?

Yes.  I have seen pet birds put their head under a wing so they could sleep when the room lights were on. 

When they do this, they are preening. Preening is when birds straighten and clean their feathers by picking out bugs, old dead feathers, etc using their beaks. They often put their head under a wing when they do this to clean underneath the wing. I have chickens so I see this every day. It's cute, although it depends on the bird I guess.

Yes it's cute af.