User Manual Renogy 3500W PURE SINE WAVE 48V Pure Sine Wave Solar Inverter & Charger

No. It would need to have an interface to your generator in order to control that other device's on/off function, which it doesn't. If you want to use a generator to charge your batteries, it would just run indefinitely, or until the gas ran out.

Yes there is a Chassis ground. If you can use a grounding rod at your location than connect the chassis ground to it and the ac out. My install was on a boat and it works great!!

Not sure about your wording. If you are wondering if this can be hooked up to shore power, absolutely! If you were hooked up already, you would already be using grid power. The batteries should never really get lower in state of charge than when you first hooked up as long as the inverter is wired properly. Hope this helps