Sounds like some sorta wah to me.
Set Up

Fender Highway One Stratocaster
Fender Princeton Chorus Amp
Fender Acoustic 12 String
Samick Acoustic 6 String
Line 6 POD X3 Live
Yea I can definetly tell that there is some wah in there. I don't really know what to tell you. I could go look in my booklet that came with the CD to see if it mentioned a brand name. I'll get back with you in a few minutes.
It doesn't say in the booklet. Sorry man. If you want, I could watch the DVD that came with it and see if it mentions anything. Give me a while though. Best of luck.
In "Closer," I'm about 90% sure it's a synth bass, meaning it was programmed. Many of NIN's basslines are like this, especially on Pretty Hate Machine and The Downward Spiral.

Live, NIN's bassist (Jeordie White at the moment), doesn't play that wah part, the keyboardist does. The bassist only plays in the chorus and bridge of the song.

The advantage to having a synth bass is that on some songs the bassist picks up a guitar and then the band can have 3 guitars while keeping the low end.

All NIN's live members (except for the drummer, AFAIK) are multi-instrumentalists.