You didn't even read my comment, and you didn't even use your brain. I'm trying to help you but your lack of reading comprehension is making me not want to help you.

Where did I say to use a different graphics card in your computer? I didn't. I said put your graphics card into another computer. Ask a friend or family member to use their computer for a few minutes to try this. All consumer grade motherboards come with 1 PCIe slot as they aren't just used for graphics cards.

Is the display cable required?

Umm... yeah it's required. How do you plan on getting a picture on your display? Of course it's going to cause a boot error, you aren't going to see anything going on with your computer if there is no picture. You can guess as to whether or not your computer is working, but if you don't see a picture then you can't know.

Edit: I sort of mentioned this in the post, but my graphics card is my previous card. It was working just fine before I switched it over.

That mean absolutely nothing. You could've killed the card by handling it. It takes as little as 100 mV (millivolts, 1000th of a volt) to kill an electronic, and we cannot feel that. You could've killed the card when taking it out of your old setup, putting it down, picking it back up or installing it into your current setup. The only way to know would be to install it into a computer and seeing if you get a picture.

In the same vein, would a broken PCIe slot cause a failure to post?

Absolutely. Computer perform a POST immediately when you press the power button. Power On Self Test, POST. The computer checks all the hardware to see if it still in working condition. If your display cable was plugged into your graphics card and either it or the slot are damaged then the card wouldn't pass the POST and you would get an error, as you already did.

The only way to know whether its the PCIe slot or the card is to test the card in another computer. If you can get a picture on that second computer using your card then the PCIe slot on your motherboard is broken and you should start the RMA process asap. If the second computer does not get a picture then your card is broken, but you likely cannot start an RMA because the card is probably no longer under warranty. If it is still under warranty, and does not give a picture on that second computer, then start the RMA asap so that you can continue to use the computer you likely spent a lot of money and time on.

Just in case you didn't know what RMA stands for, its for Return Merchandise Authorization. You cannot just return something to a manufacturer without them knowing. They would either send it back with several questions or just not return it or the replacement to you as they would have no idea why it was even sent to them in the first place.