TheFrumGuy – generally speaking, only close family remain for the mitzvah tantz, that’s no stirah to the mechitzah the rest of the time. Furthermore, in many communities, the kallah’s face is covered.
The source of the minhag is in the Machzor Vitri. But today, for all intents and purposes, it is a chassidish custom.
Gamanit, some families have the chosson and kallah with a gartel as well, although they usually fold it up so it is shorter.
OP – this is not a NY thing. Come to EY and you will see it, too. The chosson and kallah do not really dance; either they sway a bit from side to side or walk around in a circle.
BTW, as far as I remember, this is not the first thread on mitzvah tantz.