By the way, what should I expect when I write a support ticket at the AMD site? Should I get some answer? An automated email or something? A ticket number? Because a ticket without number is completely useless.
The tracker assigns a number automatically when a new bug is entered. For example, przemnet's ticket was #560.
As per recommendation from fglrx bugtracker http://ati.cchtml.com/show_bug.cgi?id=560 - BIOS upgrade on my motherboard (Gigabyte GA-MA69G-S3H) was the solution! I upgraded to the newest version which is F8E. I was able to install latest Catalyst 12.6 drivers succesfully.
I had problems with upgradeing BIOS from Windows XP and Windows 7, but upgrade directly from BIOS with Q FLASH worked perfectly! To do it, just download BIOS from Gigabyte site, extract the archive, copy BIOS file to USB stick, plug USB stick into computer, reboot system, hit "End" button during startup and select BIOS upgrade from HDD- you can find lots of videos on YouTube about that.
Many thanks for your help and support!
P.S. I wanted to edit my initial post to put solution there (to avoid going through full discussion to search the answer), but I can not do it...