Can you boot a MacBook Air from a USB 3 external hard drive?
Yes, I just got a new Mid-2012 MacBook Pro 13" and did this very thing over USB 3.0. I know of no reason why it would not work on a MacBook Air.

When booting from an external spinning hard drive over USB 3.0 on my new MacBook Pro, the external hard drive sends data back and forth to the Mac about as fast as it would if it were installed internally. I have not specifically measured drive speeds in this scenario; I'm just saying that it performs quite well.

As far as I know, all Macs with USB have been capable of booting from an external drive connected via USB--all the way back to the first iMac circa 1998, which only made use of USB 1.0. Booting from an external USB 2.0 drive, whether a flash/thumb drive or a spinning hard drive, is, however, quite slow.

When you boot your Mac, immediately after the startup chime, hold down the Option key. Your Mac will show you a row of icons of each volume or partition currently connected to the Mac (internally or externally) that holds a Mac OS X installation that you can successfully boot from. Use the arrow keys to select which one you want to boot from, and hit the Return key to boot from that volume.