This sounds like the very same problem I had last Saturday when I picked up my gtx 970.my components aren't the same but it certainly looks like you've had the same problems as me.
Mine too is a gigabyte g1 ,motherboard is gigabyte x79-up4 and my previous cards were sli 670.basically the exact same thing happened to me when I installed the 970,wouldn't post and couldn't get into the bios either.eventually took 970 out and replaced with one of the 670's and it posted and booted into windows no problem.i then tried putting the 970 in while the 670 was still in the other PCI slot rebooted and installed driver to see if it would recognise the 970.
Upon the next reboot it found the 970 but for some reason it had a yellow exclamation mark on it and said the the card would not function correctly because there was not enough free resources :/.
Strange I thought,so I rebooted and this time I tryed to enter the bios and the problem became apparent,it just wouldn't load the bios at all.so I did the old power switch trick to go to the backup bios and right away it said that my main bios was corrupted and it was installing my backup bios.groovy I thought problem solved

.
Not quite though,after it had done the backup bios it still wouldn't load the bios.so as a bit of a last throw of the dice I booted into windows and used the gigabyte flash util and updated the bios from their website.rebooted machine and hey presto bios was back and functioning
I then powered down removed the 670 and put the 970 back in the same slot and bang first time machine booted and driver installed ,970 driver installed and basically working.
Still doesn't seem perfect(it doesn't seem to like having anyone of my 670's in at the same time as the 970 for physx) but all seems right now.to be honest I'm not a fan of this motherboard and may end up swapping it out at some point.Always seemed a bit flakey to me no matter what bios version I've been running,but hey that's another story

.
So to surmise,try to flash the bios in windows with the 970 removed and either your old card or onboard video on then introduce the 970 after the flash.hopefully that will cure it.wonder if these gigabyte g1's are the cause or just unlucky coincidence??