Yes, to enable parking monitoring mode, you need a hardwire kit. This kit is not included and please purchase separately. SKU: GW41.0100. 

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 

Yes …. You can use anywhere it goes with you… I use mine in our camper 

The European Standard Type 2 charger is suitable for most electric vehicles in Europe and other countries because it meets the European Standard Type 2 standard. You need to determine if your vehicle is equipped with a Type 2 charging interface in order to use this charger. 

I've pretty much only used these batteries when doing video work, and haven't gotten any hum. Hope this helps.