You can run all your gear while the battery is charging because as someone already said the battery will act as a buffer In just the same way it does when you plug something in to a Cigar lighter socket, The One thing you must never do is run your Fridge directly from a Petrol/Gas powered generator, Because when they are about to run out they rev up and send a voltage spike straight in to your fridge,
If you are using a generator to charge your batteries via it's built in charger you can still run everything including your fridge as long as everything is connected to the battery first, and you can also run everything if they are connected to the Battery first "and you are using a Proper Battery charger that is being powered by the generator, But Never power your fridge via the 12v supply or the 110-240v supply on the generator because if it Revs up it will kill your fridge, It might be ok if you have done it a few times But Doing this is the biggest Killer with 4x4 fridges, "ALL" Power to a Fridge ( MUST ) come via Battery with 12v power or by using a 12v powered 110-240v Inverter.
Using your gear while charging is the same as using your gear when you are using solar panels to top up your batteries,
Hope that helps.