Storing my iTunes library on an external flash drive in a MacBook Air
Has anybody done this?


Not with a flash drive, but with an external drive. I've kept my iTunes library on a USB-attached drive for years now. Semantically, the situation is identical to storing it on the flash drive. The OS see's a remote disk in both cases.


  Any advice?


Format it as Mac OS Extended (Journaled). Most flash media come pre-formatted as FAT32 these days. You don't want that.

For moving your iTunes library the correct way see: What is the safest way to move the iTunes Library from one hard drive to another?


  Would it be backed up with Time Machine correctly? 


As long as it's not listed in the Exclusions list in Time Machine, yes: it will be backed up. To check the exclusions list: System Preferences -> Time Machine. Click on the Options... button and make sure your flash drive volume isn't listed in that list. If it it, select it, and click the - (minus) button at the bottom of the list to remove it.