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A car battery shouldn't be discharged at all. They are meant to supply a short burst of power and then be charged back up again.
A deep cycle lead acid battery can be used to run an inverter. They can be discharged to 50% capacity while still providing hundreds of cycles. If they are used for a backup and are not cycled frequently, they can be discharged to around 80%, but they will provide a lot less cycles.
A LiFePO4 battery is definitely the better choice for anything that needs to be cycled frequently though.
Just wire the inverter to your car battery and run the engine. Hard on gas compared to a 3500 watt generator, but you already have the car, inverters a few hundred bucks, and the genny would be at least 2500 dollars
An alternator can't output full power when the engine is idling. You may only get a few hundred watts before the battery starts draining. You can get a generator that will produce much more power than a typical car alternator for under $400.
You're speaking like an alternator is a generator, and doesn't come in various outputs. Your average larger vehicle can charge itself from near-dead at idle or run a 3500 watt inverter, although I have had the odd small car or motorcycle incapable of either.
Sure but in an emergency? They can handle being discharged as long as you don't go too far.