They do, stand corrected.
Don't get the 2,000. It likely will not run any A/C.
Most any Air Conditioner will need that 2,200.
Don't confuse AC Air Conditioner with ac Alternating Current.
120 is current I assume you mean, yes, all Air Cond units will run on household current like generator puts out.
If running a fridge too, a 120v household current fridge, then these little generators will not suffice. The air cond will be maxing out the generator at start-up, so if you want to add a fridge then get something in the 3,000 area. Fridge has the same type of compressor, so needs start-up power, and no way to tell them to never start up at the same time.
Good luck with solar. Do you have lots of panels? I have 200 watts solar and this is enough to keep things running, but not an inverter for long, and def not an inverter to run a lot of things like you want to. I use the inverter to charge the lap top on rare occasion, otherwise it always stays off. The more things you can make run straight off the 12 volt DC the better. The same appliance running off DC will use way less power than its household current version running off the AC (alternating current) that the inverter puts out.
I have 2 batteries and run an ARB DC powered fridge, lights, a fan, water pump, maybe the furnace fan and charge the phone, but little else - not all at the same time though - all this runs off DC. On rainy weeks I can last 2-3 days on what the battery has. Very little solar action happens in the rain. You also have the lift pump to raise the roof which will use a lot of power, so to start the camping off by draining that amount of power down is not ideal. I have separate battery for camper and for pump. You can run the vehicle engine while you raise the roof, but there just is not enough power over the trailer plug to matter much in this instance.
happy trails