AIR iOS - Check if app was previously installed?

Actually you can develop simple Adobe AIR Native Extension (ANE) that will utilize keychain in your app. As different sources telling - keychain could be cleared only after user made "Reset All Content and Settings". Btw, "Reset All Settings" will not clear keychain.

1) So you must write to keychain some data at the first launch

2) After app was removed - keychain will hold your data

3) After app was installed again - you can check device keychain and will know that was installed early or not (sure if user don't made content reset

I think not usually, no. You could have a server side database that the app would call to when it was first used. One problem is that you can't easily find out a unique identification for the device, to know that the user had reinstalled under the same device.

If you're wanting to reinstate game progress, maybe one of the game centers would help solve that?