What happens if you just turn on airplane mode on the phone and try to use the adapter? If it's really creating an ad-hoc network to do this, the output shouldn't work. Disappointing to hear about the quality of the image, regardless of how it gets there. Apple, you can and have done better.

Press the power button on the top, it should go right to sleep. You don't really power it down much, but if you needed to then hold that button down for about 10 seconds, it'll ask you to confirm and then actually shut off. There is a built-in speaker, but it has a low max volume and is basically not useful for much besides hearing alarms. Watching movies or listening to music through it is painful.

The Air does not have USB target disk mode. You're confusing it with Remote Disc. You can verify this by either (a) holding down "T" at startup (which it will ignore), or (2) going to the Startup Disk prefpane and seeing that the option to restart in TDM is gone.