It does but it's not reliable. The update process is also the slowest imaginable other than Windows update. It takes several minutes at times just to discover which games need updates.

If you haven't played a game recently it will punt it out of the auto update queue.

It probably does, but it's turned off the majority of time. It's still plugged in, so I wouldn't mind if it updated itself silently, but no, it does not do it when it's turned off.