Shorepower

It's my first semi truck so sorry if this is already common knowledge and hopefully its the correct sub for these questions.

I have a plug on the side of the truck. Truck came with xantrex freedom hf 1000 inverter/charger that I understand is stock from the factory.

If I run a generator to that plug will that charge the batteries and provide power to the outlet sockets?

If I upgrade the inverter and get a 3000 watt generator would I be able to plug in an 8,000 btu portable air conditioner? Is that how this would work?

What are good brands for a 3000 watt inverter/charger?

 

Closest thing I can find online shows block heater by the engine bay and separate shore power inlet on cab.

 

So just plug it in and just see which sockets work with inverter off?