A portable generator will run stuff off of extension cords, unless you've made the appropriate changes to tie it into your house wiring (see below).
If properly connected to the house through a transfer switch of some kind, you could theoretically run your central air system if the generator has enough power output.
Now, properly and safely connecting your generator to the house requires some kind of power inlet and either a dedicated transfer switch, or an interlock installed in your breaker panel. This will prevent the possibility of your generator trying to back-feed through the mains and injuring electrical workers if you leave a breaker in the wrong position. Unless you're familiar with doing residential wiring you'd best hire that installation out to a licensed electrician.
The other big no-no is building a "suicide cord" with male plugs on one end that connect to the generator, and a male plug on the other end that connects to something like a dryer or range plug. Having the exposed prongs on both ends is a big safety issue--for example, making or possessing one at my employer is grounds for termination.
In the timeframe we're talking about you'd be better off with extra extension cords and a window unit or two; doubt you'll be able to get an electrician in to do all that on short notice. One of these days I'll get around to putting the transfer switch in for mine, but in the meantime a couple of good long cords will run the fridge and an air conditioner. But anything past Cat 1 we're going to stay with our parents near Atlanta anyway.
I couldn't tell you how much gas you'd need. You'll need to look up the fuel consumption for your specific generator, and to be conservative assume full power output. Rectal-extraction figure says a little north of 100 gallons.