can you use airplay mirroring to extend the desktop from your MacBook so that for example you can display a video on one screen and perform other operations on your mac? This setting is available when using duel displays through thunderbolt/hdmi cables.

Yes, you can. XD

I thought now airplay is eqully functioning as HDMI/thunderbolt cables, wirelessly. But only one problem, with 1080P videos, it is a little bit laggy for my Macbook Air 2011, perhaps if you get a newer model, this won't happen at all.

I just tested it again, with youtube 1080P MTV, the image lags a little bit. I have not tested with HDMI connecting. I can not be sure weather it is the problem with Apple TV or my Macbook air 2011. But playing 720p or lower defi video, it works pretty good.