If both PCIE power connectors were not in, it would probably say something along the lines of "Video card not getting enough power, check that all the power cables are plugged in" or something like that.
Check in the BIOS to see if there is a default video out port. If there is one and it is not set to auto detect, try to set it to do that. You would have to do this while the video card was out and the setting might revert when you plug one in.
For good measure, you could try taking the battery out for 30 min. It is a CR 2032 watch battery, it looks like a quarter.
Regardless what you do, I don't think it will help to plug the monitor cable into a motherboard port if the video card is plugged in, so I wouldn't bother doing that. At best you would be operating without the use of your video card if you did get it to work in the motherboard port.
Just to prepare you for the idea now, Zotac isn't the greatest brand. The video card may just be a plain DOA. If you have another computer laying around, I would try to use the Zotac video card in that and see what happens.