The Honda EU 2000 is a good unit. I have one, it's quiet, and can generate power for alot of your needs. however, it won't have enough amps to start your Air conditioner. you would need two connected together to do that. a EU 3000 would be good. Heavier and louder though. calculate your amps your gonna use and see what you really need.
You say you got one battery. get a second one and hook it up in parallel. This will increase how long you have in reserve when the generator is off. Some places you go may have quiet hours, or generator hours. or you may have neighbor campers not wanting to hear a generator late at night. One battery may not last all night. and you don't want a battery to go less than 50% if you can help it. These deep cell batteries last longer if you don't drain all the way down. a heater fan can easily drain your battery on a cold night.
Another thing you can do to save amps id to replace your incandescent light bulbs with LED. a 2 amp incandescent bulb can be replaced by a .2 LEDs, giving off the same light. look around the web, and see the reviews and such for the correct bulbs. a bit more expensive, but you can find deals. well worth it on amp savings. you can run 10 leds on the same amperage as one incandescent. or one, 10 times longer.
To bump what Sarge said about the heater... we experienced that "once"... when the power is lost the warning stcreach comes on which makes your ears bleed... it is uncontrollable unless you unhook your battery all together which I had to do in the middle ofa winters night... burrrrrrrrrrr