I'm considering switching to developing on Flash Builder 4.5 since apps can be built for most platforms. So would it be possible to include Adobe AIR with the app or somehow have it hardcoded into the app to automatically download Adobe AIR if they don't already have it without prompting the user?
You do know that Adobe discontinued development of Flash on Mobile.
E.g. read
http://www.guardian.co.uk/technology/2011/nov/09/adobe-flash-mobile-dead
http://www.wired.com/gadgetlab/2011/11/adobe-kills-mobile-flash/
http://www.engadget.com/2011/11/09/adobe-confirms-flash-player-is-dead-for-mobile-devices/
Adobe does not say this officially but without a mobile game for Flash they basically admit that Flash is dead and HTML 5 is the future. If they don't bank on it, nor should you.
Update: It seems that Air is still being developed by Adobe to run on mobile devices and is not dead. It might be a suitable platform for mobile app development but I am not qualified to judge on that. Personally I lean towards truly native apps but depending on your business case for a specific apps other approaches might be suitable.