The card is working in the basic sense its allowing your PC to boot up. So its posting as good. but if the outputs on the card are borked you are never going to get any sort of display out.
Teamviewer also displays your desktop via software. Not through output on your video card. (hence why if your outputs are bad on the video card you wouldn't get anything displayed on your displays)
But I mean everything you can do you've done.
Different outputs, different displays, different cables. (and i imagine reinstalling drivers by now). Thats all you can do. You can swap PCI-E slots as well.
If your not getting any display when you turn the PC on to boot, means your not getting any output period. Which means bad outputs on the card which rarely happens but I've seen it happen here a few times.
I5 3570K@4~5.0ghz: Gigabyte GA-Z77X-UD3H:Corsair Hydro H110 :16GB Corsair Vengeance DDR-3 1600: XFX Radeon 7950 @ 1100/1575 : Corsair AX 860 PSU: x1 WD Black 1TB HDD's: 90GB Agility 3: 960GB SSD : Creative Labs X-FI:Corsair 500R Arctic White Edition:3 22inch Monitors Eyefinity @ 5294x 1050 : Alienware Tact-X Keyboard: Razer Naga :Corsair Vengence 1500 Headset: Corsair SP2500 Speakers