No. It’s over the ears. 

So when it powers off, it goes into a sleep mode. That sleep mode doesn't require alot of power, but if you have it set to turn on if the camera detects vibrations, it uses the cars battery. Mine is plugged in all the time and it only powers on when I turn the car on or it feels vibrations. With the vibration setting, it will turn on when I open my car door. But if your power source isn't powered when the car shuts off, then it doesn't use any power, just what the camera has stored up. Hope that helps 

I haven’t used it for sports but it has a pretty fast zoom, so I would assume that itv would do just fine. 

Yes, there is a metal piece at the bottom center that you can probably hang a weight from.  

Yes, it can be automatically paired after being taken out from the case.