Not sure where you read that - air con uses power, so turning it on
will make the revs drop. That said, it shouldn't make the car get near stalling - perhaps your idle speed needs adjusting.

To use: Turn on air con, set temperature dial to cold (blue). That's it.
'Circular arrow'? Assuming you mean this is a button, it's probably the setting to recycle the air inside the car rather than get it from outside all the time (since it's already working with the cooler air-conned air in the car, it's got less work to do that way).
Using
any accessories in the car diverts engine power for the purpose, but air con is particularly bad for this. Not a huge difference, but quite a big one really (most accessories use so little really as to be pretty much un-noticable). Running with air con all the time will probably reduce your mileage from a tank by 20 or 30 miles or more (educated guess - someone please correct me if I'm wrong here!).
As to why your wife says don't open the windows - I've no idea, unless it's p!ssing down with rain or something, in which case she has a point

Or If the aircon's on, then it's a bit pointless to have the windows open too... although that could be you doing your bit to reverse global warming...
On the subject of windows - opening windows will make your car cut a less clean path through the air, leading to more petrol being used, but a car with all the windows down will still get more miles from a tank than the same car with the windows up and aircon on.
Personally I don't like Aircon. I don't like the way it dries the air out - tends to de-moisturise me! And although I don't actually have aircon in my golf, even with hire cars I'll tend to have the windows open and aircon off as long as I'm moving, if I've not got an aircon fan in the car with me.