I'm getting a 360 Elite next week (it's en route to my house). Which means it has an HDMI cord packaged with it.
My notebook has an HDMI input. If I connected the cord to it, would it output on the screen? And if so, would there be any way for me to take screen captures, or is it "disconnected" from the OS while it's doing that?
no, your laptop would have to have ViVo capabilities (Video-in/Video-out) which are rare even on high end desktop computers/graphics card. So realistically there's no way to "plug and play" your X360 to your laptop and actually get image out of it as it is certainly only a Video-In one.
You're gonna have to settle with either a USB video adapter/capture or with a pcmcia/express card who does video capture, both kinda expensive and both mild prone to some lag (specially in FPS's/fighters and action games); not to mention I don't know if is there expresscards capable of capturing HD resolutions in real-time already.