Hi there,
Like most motherboard manufacturers Gigabyte doesn't offer updates or drivers for Linux based distributions and points user's to other sources for these. Updating the BIOS might help with your shutdown problems but the only way of knowing for sure is to try it and see.
When a BIOS is shown as a BETA version in the download list it is usually solid and there are many instances where a BETA BIOS is never updated further so there is normally no problems when running one. Of course, one advantage of using QFlash to update your BIOS is that you can also go backwards, to an earlier version, if you so desire.
I think, if I were facing your situation I would update to the next BIOS, F3 and see if that solves the problem. If it doesn't I would then go to F4 and if still no luck then F5a. I'm afraid it is really a case of suck it and see and of course, the problem might not actually be with the hardware but with the software.