Yes, you can use 4 or 8 rechargeable AA batteries as backup. 

This camera has a large 15,000mAh battery, and it keeps itself charged with 2 methods: solar power and power sources. On cloudy days, the camera relies on its battery. However, heavy camera usage might eventually drain the battery. You'll receive a notification if the battery level gets too low. 

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 camera does not have an on/off switch. If you want to switch them off completely, you have to disconnect them from the power supply (e.g. when not in use) but you can also leave them plugged in.