If your TV has a USB port (mine does, and is in no way state of the art) you should be able to power it from that. Not quite a zero cable solution, but it's close.
PoE explicitly only works over Cat5+ cable (although I think there's a low-power version for Cat3). HDMI obviously doesn't have the same physical-layer properties as Cat5, even if you're passing the same Ethernet frames on top.
I dub it the "reach around pc" when it is plugged into TV on both ends, otherwise it's just like a guru/shiva/dream plug.
http://www.globalscaletechnologies.com
I had dreams of just plugging this stick into an HDMI port (TV, monitor, whatever) and having an instant PC. Sadly, it needs external power.