Wow, kind of a older topic.... anyway....
I wouldn't be concerned about the AC unit running during a dust storm, except maybe when the storm is over clean off the fins on the roof. Easy to do, but be careful because the fins are really thin metal and will cut you pretty good.
My concern would be with the generator air filter. Should be fine.
For sure go out and pull the air filter off and clean it after the dust storm.
But if it is clogged during the storm, I'd imagine you could try cleaning it. Gear up, grab a dirty sock or something to plug the hose into the motor while you have the filter out. Take the filter off, run in side clean it best you can, then out out and put it back in. Don't forget to plug the whole while you have the filter out, and to remove the plug before you put the filter back in.
Why don't ya stick your head in that hole and find out? ~piehole
Plan for the worst, expect the best. Make the most out of it under any conditions. If you cannot do that you will never enjoy yourself. ~CrispyDave