Hopefully you can find a PCI card that will work, any should be fine. Then we should be able to quickly rule that out as part of the issue or not.
As for your memtest issue, if the new one does the same then we obviously wont be able to adjust anything until we figure out the cause first.
So... For now lets do this quick memory test. Test each module, one by one, in any white slot and see if the board will boot and run, and stay running longer in the BIOS than before. So I'd say just test each one, one at a time in any white slot, and enter the BIOS and sit there for 3-5 minutes and see if it stays running. If so, move on to testing the next stick, this will rule out any one stick as being possibly faulty and causing these issues when you use the whole set.
Yes, it's normal to shut down after making BIOS changes, sometimes it will fully power down, other times only partially depending on what you are changing. Nest time you can get in there please try to load optimized defaults as quick as you can and then hit F10 to save/apply/reboot.
Some USB keyboards and or mice remain powered up while the board is shut off, so that is normal, PS/2 as well sometimes depending on the BIOS settings.
Memory must be in the white slots only for 3 modules or less. So please number each module with a sticky note, or some tape and paper, and see if you can do the above test or not.
7F error is noted above, mainly it means that settings failed and fail safe defaults are loaded, which is why I mentioned for you to try to load optimized defaults as soon as you can. If you do end up being able to, once you reboot go back into the BIOS and set 1.66V ram voltage, this setting is in the MIT section, then into sub-section 4 at the bottom of that section.
I'm still on the fence as possible graphic issue, but it could be memory related as well (Settings or faulty dimms)
Do you have a case speaker connected? If so, do you hear any beeps with you see 2b on the debug LED?