I've only played the DC version and the Japanese NESTS collection on PS2. Both of those feature the extra backgrounds, so I can't imagine any of the versions would lack them.
The special stages only appear in versus mode and are triggered by strikers. For example, the normal roster of strikers gets you normal 2000 stages. If you or your opponent choose Kaede, you get a Last Blade background. King Lion gets you a Kizuna background. Fio results in a Metal Slug background. It's a bit random, though. It's sometimes based on player 1 and sometimes on player 2.
Don't know of any videos of these stages, since most KOF2000 videos are from either arcades or emulators.
I'd recommend getting the PS2 collection if you can play Japanese games. The US version was apparently censored, although I'm sure it plays fine. Also, the collection has color edit, if you enjoy that.