Do you need a minecraft account to use the Xbox 360 version?

I was thinking about buying the Xbox 360 version of Minecraft and was wondering if I also needed to buy an account to play offline.

Added (1). I also started wondering if I needed to play in HD. My TV is pretty old and I'm still using the video and audio cables. Will this work?

Nope.
Once you buy the console version, you can use however you want. No need to have a minecraft account.

No you don't need a minecraft account to play it on the 360. You just need the disc you buy (or the download version from the live game store).