Yes, you can run it with an external monitor as the exclusive display, but you'll need to reboot in that state.
If you need to switch in and out of that state while the notebook is on, then you'll need a tool that prevents sleep - there are a few about - and a willingness to tolerate the 'find your cursor' game when your pointer goes offscreen into your MBA's monitor.
Much less flexible than a Vista'd ultraportable, but hey - you move down, you have to put up with some inconveniences.
I don't know about the new displays so can't tell you. The old ones won't for sure - they even have trouble powering regular USB stuff at times