Or buy wireless x360 controller, start up steam in full screen mode, and get 10meter hdmi cabel to connect it with tv in other room, and you can cotroll your pc with that wireless controller.
Then get some good Cat6 or Cat6a cable and run it through that, just remember there's a maximum distance of 100 meters for ethernet cabling. That would be the easiest way to run the cable. T
I have no answer to your original question though.
Could it? I'm sure it could, would it work as intended, well we get into a grey area there. I used a similar device to that one shown to run a ~250 foot stretch to connect my churches projector to the computer controlling the program and it runs fabulously at 1080p with no signal distortion using Cat6, cat 5e didn't cut it though.
With that said, again, I believe it would work fine but there's no way to be sure.
Interesting stuff. I was looking to do a similar thing, but I might just wait it out and see how streaming on Steam OS turns out and perhaps stick it on an old system