Might feel like a step backwards in terms of environmental impact, but presumable the fuel used would be minimal while the sun was shining or batteries not empty.
Yes.
Our neighbor across the street depends on a respirator. We live in a rural Bay area location that has grid dropouts several times a year. Some last a few seconds, some more than a day.
His setup: A transfer switch operates ~ 3 seconds after grid failure, followed ~ 2 seconds later by autostart of a propane generator. His house stays on generator power until the grid comes back up for at least 5 continuous minutes. Then the transfer switch flips back to the grid and the generator shuts off.
Why propane? Because gasoline slowly degrades (polymerization) and needs to be drained, disposed of, and replaced with new (typically every year or so). Propane doesn't degrade over time.
If you have natural gas piped to your home then a natural gas generator is a solid option. Expensive, though.
With the normal grid-tie systems, all of the energy produced by the panels must be consumed. If you are producing more solar energy than you are consuming, while running on generator, the inverter will still attempt to force that energy into your home. The generator will take the brunt of it and will likely be damaged or destroyed.
They do have systems that can take inputs from Solar, Generator, Grid, and Battery and use the best available option. But they are significantly more expensive than the typical grid-tie systems.