The phone app is just a GUI frontend to the ASIAir software running on the Pi, it can't do anything standalone apart from maybe run the sky atlas. Once you start an imaging session you can close the app or turn off your phone and the Pi will carry on doing its stuff.
There is a mount simulator on the Air, it's listed as Demo Interface in the list of mount options. It's part of INDI which is open source software. INDI also includes a CCD simulator but ZWO don't seem keen on allowing you to use it, I've requested it a few times but never got a reply. A CCD simulator would allow you to test/learn the full functionally of the Air, including plate solving, PA and even guiding.
