You're safe. This product is UL listed, and the "wire in the wall" is part of the product itself. The wire in the kit is probably higher quality than the Romex in your house!
You plug your own power cord into the bottom receptacle, and plug your TV / Cable box into the top. NOTE: you'll need a small power splitter, or multi-socket pigtail at the top, unless you're only plugging in one thing. In my case, I needed power for TV, FiOS box and DVD, so I got a 3-way splitter.
I would highly recommend this kit.

This in fact is code. You just have to make sure any other wires you hide behind the wall are rated that way too. Not all HDMI cords are coded to be put behind the wall.

Wires can run behind walls if shielded properly. Its the way new homes are built.

I cannot answer whether an electrician or inspector would use or approve, as you know codes and personal preference varies. I was satisfied with the quality of the connectors, gauge of wire, reputation of Wiremold solutions and it is UL approved. The cable is rated far above the current load of a television. The alternative for a finished home is to have an outlet added, move an existing outlet or use the provided TV cord and route through the wall. This was a valid alternative in my personal opinion.

The wires in your home already behind the closed wall? I don't see the difference.