Ok, I just booted up my other computer and got into the BIOS. Under advanced BIOS Features, try changing the CPU core control to manual (from) auto, to see what happens. From the screenshot you posted of the cpu's, it is different than mine. Rather than group Cpu core 0 and 1, 2 and 3 together, I see all 6 cores (in my case) individually listed.
I can't imagine why @BIOS won't let you flash to F3. The other method (Q-Flash) may work. To use that I think you have to first download version F3 from the gigabyte site and then hit F8 on the main BIOS screen. I haven't used that method, so I don't want to make things worse for you. Maybe someone else can offer advice on that method.
But you definitely (per gigabyte) need at least F3...