Yes it does... I play on an old TV in my room all the time with it using the Wii cables and just plug it into the back of the WiiU
Considering the fact that he is running just a GTX 580 I doubt he has the specs to be able to render and upload video at 1080p in a reasonable time so 480p would be the best choice for him. The Elgato products don't have game mode that would allow recording and playing on the same screen unless you can deal with 5 second lag between them. Their products are made to passthrough to a TV and using the computer just to record.
http://techforums.nintendo.com/thread/17846