I don't think it would as the keyboard uses bluetooth and Apple TV doesn't support that, to my knowledge. If you have an iPhone, iPod Touch, or iPad you can use the Remote App to pull up a keyboard that works with Apple TV.
I don't know. I think you have to use the remote that came with the Apple TV or you can pair it with a remote that can control multiple devices. I am not sure if an Apple Keyboard will work with Apple TV