Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
Origins of the Sun Dance. According to historians, Sun Dance preparation amongst most of the Plains peoples involved a lot of prayer, followed by the ceremonial felling of a tree, which was then painted and erected at the dancing ground. All of this was done under the supervision of the tribe's shaman.