I also couldn't get Adobe Flash to install on a new Windows 7 computer. At first, I got an "unable to register" error. I followed the instructions posted here to install and run Microsoft's SubInACL program, and that error went away but it still wouldn't install. I tried uninstalling and reinstalling multiple times (including running the installer with Administrator privileges), and tried installing from IE and FireFox. It would say it installed successfully, but still wouldn't run. When I'd visit the Adobe's web site it would say I still needed to install Flash.
I submitted a trouble ticket to Adobe and got a quick response. They had me do the following:
In Internet Explorer 8 click on Tools then Internet options then the Advanced tab then choose "Reset Internet Explorer settings".
Download and run install_flash_player_10_active_x.exe from Adobe
Navigate to C:\Windows\System32\Macromed\Flash (may be a different path if you're on the 64-bit version)
Right-click FlashUtil10d.exe (you might have a different version), click the Compatibility tab and select "Run as Administrator." Click OK and run the program.
That did it. The difference must have been the first step - resetting IE. I had tried all the other steps repeatedly.