I got it to work. I uninstalled the driver I had and installed the full "toolkit": http://forum.xda-developers.com/showthread.php?t=1995688
Even though the driver I got was supposedly the exact same driver that's in the toolkit, the tookit definitely did more stuff which got my nexus 4 working -- the nexus 4 displayed a message about connecting to the PC, which it hadn't done before.
Now after going through all that, Flash CC launches the AIR app just fine.
Cheers!
-Aaron