Alright, after hours spent tracking this problem and testing I got it nicely solved. Thank god I had clean install of Windows, else it would be a real damn pain to figure out where to even start. Anyway, issues were caused by AtiTrayTools, by its OnScreen Display feature to be exact. It somehow conflicts with both - Flash Player installation process and Flash Player in use. I basically created new user account and did some testing step by step. In short - disabling OSD feature in AtiTrayTools solved all problems (requires either reloging to your Windows account or rebooting the system after). I even reinstalled both FP plugins, everything went fine. I still find it pretty odd, as I'm sure I had OSD feature running once in a while on my old OS installation and didn't have any issues like these. But on the other hand, it's possible it was due to some other system settings, that prevented conflicts to occur.
Sympthoms: Flash Player fails to install, at about 50% returning "general installation error". Browser crashes when loading Flash Player videos.
Cause: AtiTrayTools OSD feature.
Solution: In ATT go to "Tools & Options" --> "OnScreen Display", uncheck "Enable OnScreen Display", press "Ok", relog to your Windows account or reboot the system.
Hopefully it will help people having similar issues in the future. Adobe's staff might want to add it to FAQ as well.
Excellent, thanks for posting back with this solution! I'll make note of this in case I run into this in the future.