New info: I was stuck between a rock and a hard place, but no longer. My Adobe Air app wouldn't install because it found nothing to process ".air" files. I couldn't install Adobe Air again because Windows 8.1 thought the current version was already installed. I couldn't un-install Adobe Air because it would exit with an error "Invalid GUID 'null' " in the log file.
Breakthrough: Today I created a new user account to try that. When I tried to install Adobe Air, a new version 16.0.0.245 was available on the app server. Because it was a NEW VERSION of Adobe Air it was able to be installed over the old version as an update. Then the Adobe Air app installed correctly and it now runs again under any user.
Conclusion: I believe there is a bug in Adobe Air when you try to un-install it from the Windows 8.1 control program. IT won't let you un-install. But if a new version of Adobe Air comes along you can update to it, installing over the old version.