Hi, it's only for recharging the Wave 2 battery, which is optional. And frankly, at 12v, recharging must take a week...

Their internet site (US) has a picture of it being used in a car and apparently charged from the car's 12V system. There is a symbol of the car cigarette 12V plug on the picture. However, the A/C is shown with its battery, so I think the 12V only charges the battery, you probably cannot run the device directly off the car's 12V supply, I don' think it has enough power (although that varies from car to car). You don't get the 12V cable with the A/C, and I also didn't get it with the battery, which I bought separately.

I don't know if this is a good idea. There should be 10A permanently available at 220V, the cigarette lighter will probably be fused with max. 2A. If it works, you risk damaging the device.

Yes, but it uses a lot of energy and the cable gets very hot.

Hello!
Only indirectly via the additional battery or an Ecoflow Powerstation, which you charge while riding. If you take THE battery with you or already have it anyway, take a look at the Ecoflow Alternator, which is just coming onto the market. This would give you enough juice to run it for a long time.