Yes, you can charge the animal camera with an external power supply. For the DC connection you need a 6V/1.5A adapter with 3.5mm (outside) and 1.3mm (inside) plug. 

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 

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Hello, the camera is equipped with a flash. When the lighting conditions are insufficient, you can turn on the flash in the settings and take photos. 

No, only continuous light.