Hi
For live shows you have to use the USB connection, that's how your camcorded functions as a webcam, so make sure the one you buy does have a USB port (well, almost every model does).
For recording forget about DVD and I'm not sure about you needing HD resolution, so choose from the many MiniDV or HDD cameras. I have some experience with Panasonic and Sony camcorders used for live streaming video shows, both work perfectly. However I'm a Canon fan, but if you had some bad experiences with it, Pana or Sony is the way for you.

Good luck!