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

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 

Our wireless endoscopic cameras work with cars/vehicles. Such as carbon deposits on piping valves, removing missing bolts in engine bays, inspecting cylinder walls and pistons, gearboxes, muffler pipes, etc. 

Yes, it works with any camera as long as the camera has a external mic input. But you need to set the output volume to low levels, and listen on the headphones jack of the A7s to make sure you don't overmodulate the audio into the camera. The Sony takes only mic level signal.