Can an Xbox 360 be hooked up to a Thunderbolt display? Having both the video and audio going to the display would be optimal, but if I can just get the video on the display and use headphones plugged into the Xbox, that's perfectly fine and probably easier to achieve.

I have an HDMI to mini DisplayPort adapter, so I tried using that and plugging it into the Thunderbolt port, unfortunately I didn't get sound or video that way.
The Thunderbolt display expects a full Thunderbolt signal (that is, muxed PCI-Express and DisplayPort). It will not work with DisplayPort-only sources.

To date, I've not heard of any announced (let alone released) adapters that could drive a Thunderbolt display from a non-Thunderbolt source.

If you really need to do this, you might be able to hook the Xbox up to an HDMI capture card (it needs to support HDCP) that is connected to a Thunderbolt-equipped computer to which the display is connected. These capture cards are usually used by professional video editors and can be quite expensive.