Do you see any messages for the location of crashlogs while installing AdobeAIRInstaller.bin in silent mode from the terminal?
$ ./AdobeAIRInstaller.bin -silent
As you have tried earlier versions of AIR (alpha/beta), so that maybe causing the problem with installation of latest runtime.
Please read here: http://blogs.adobe.com/air/2008/12/tips_on_resolving_application.html
and it might help you in resolving your installation issues.
Also try these steps on terminal:
1. touch ~/.airinstall.log
2. chmod +x <path-of-AdobeAIRInstaller.bin>
4. Continue with the installation till it fails
open the log file (~/.airinstall.log) and send its contents to firstname.lastname@example.org
this will help us in resolving the issue