I would like to connect a professional USB 2.0 camera to the Stick's USB port (2.0 is fine) and have it displayed out the HDMI port.
Probably use AMcap (that works on my Windows Vista) or maybe a more sophisticated video & capture program.
Streaming the video out would be nice too but not at all required for now.
The Compute stick has the basic functionalities of a normal computer but this setup you are requesting will depend on the drivers and design of the camera.
If the camera comes with drivers that allow you to use it as a video input device then it may be possible but you need to check with the Camera manufacturer if it can be used for this purpose.