Living in this region myself I have no doubt the expats are safe in Doha, as with the other GCC states. With the amount of Military and Civil Police presence (especially now) this will continue to be the case. And as for the Coup stories...they are just another part of Middle East life and you get used to all the rumours.
What will be entertaining is the Aussie Hornet Sqn and their support personel deployed to Qatar will receive medals and money for being in a "war zone" which we call on a daily basis "home", if they end up overhead Iraq on an hourly basis then deservidly so, but I wonder if the Aussie and Kiwi Qatar Airways pilots will earn a gong.....the tax free money they already have