Connect camera via firewire cable. Open capture utility in Premiere. The live feed from the DVX will show up in the capture window. Use Premiere's record button to capture footage. I do this quite a bit when I'm testing and it works great. It's like a P2 card, a giant, heavy, expensive, P2 card.
No, not all cams do but the DVX continues to send the live video image out the firewire even without a tape or even while you changing tapes. You might also check out a program called DVrack. It's designed exactly for what you're wanting to do plus it has various meters that will help you record better video. I think they still have a demo version you can download. www.seriousmagic.com
I also have DVrack. It works the same as the live capture in Premiere but with all the scopes and meters that TeleDan mentioned. If the DVX is connected to the computer and turned on, it's sending the live feed to the computer. So as soon as I open Dvrack or Premiere, I can capture footage.
Another fun bit about the DVX...I use a cheap cam as a capture deck. I connect the cheap cam to the DVX via firewire and set the DVX to control the other camera. I put the tape in the cheap camera and whenever I press record on the DVX it activates the cheap camera to record. I think I might be babying my DVX too much.