Yes ofc it will, I have it the other way around, an Asus Gpu in a Gigabyte mb 
Asus motherboard + Gigabyte GPU
Yes the GTX 670 will work on that Asus mainboard just fine. Just remember to get the recommend power that the GTX 670 needs. http://www.geforce.com/hardware/desktop-gpus/geforce-gtx-670/specifications
Yes, you can run it at any standard 4:3 or 16:9 rez I am aware of.
Yes it will.
I run a Asus P9x79 Pro motherboard and a Gigabyte 560ti SOC edition GPU.
3930k 4.6ghz | NH-D14 Cooler | P9x79 Pro MB | 16gb 1866mhz G.Skill | 128gb SSD + 2×500gb HDD
EVGA GTX 780 Classified w/ EK block | XSPC D5 Photon 270 Res/Pump | NexXxos Monsta 240 Rad
CM Storm Stryker case | Seasonic 1000W PSU | Asux Xonar D2X & Logitech Z5500 Sound system |
EVGA GTX 780 Classified w/ EK block | XSPC D5 Photon 270 Res/Pump | NexXxos Monsta 240 Rad
CM Storm Stryker case | Seasonic 1000W PSU | Asux Xonar D2X & Logitech Z5500 Sound system |