(Below the line is the original text of this OP, which I thought was too wordy and have rewritten for clarity.)
My GPU, when connected to the power supply, causes the computer to not boot. It will instead power up all the fans (PSU, case, CPU, GPU) and then turn off after about 20 seconds, then turn on again after 5 seconds, and repeat. This is the same behavior as when there is no RAM installed.
When it is in the PCI slot but without it's power connector, the fan spins even faster, but it is not recognized by the computer even after installing its driver from the disk.
I've been having some weird issues with this build from the beginning, including something where, if two RAM sticks were connected, it wouldn't boot but instead turn on and off over and over again (which is the same behavior as when there's NO RAM...), but work perfectly fine with only one stick, but THAT seems to have gone away on its own and now both sticks are working.
But now, things seem okay but I want to check this out with you guys before I do anything else. I put my GPU into a PCI slot, and saw a 6-pin power connector on the top of it. My PSU had a matching 6-pin connector that said "PCI", so I plugged it in. With this setup, the computer would do the same thing as before: turn on and off by itself without anything ever showing up on the screen. The GPU's fan WAS spinning.
Then, since I realized that nowhere in the GPU's installation guide does it even mention the power connector, I decided to unplug it and turn the computer on. The GPU's fan immediately started spinning much faster than it was before, and the computer booted normally. I haven't plugged the monitor into the GPU yet, it's still in the motherboard's VGA.
So... was I supposed to use that power connector or not? If not, why does it exist? Should I go ahead and install the drivers now?
If your using a discrete GPU (like you are) you need to plug your monitor(s) into that, not the onboard graphics you'll need to get an adapter if you want to use a VGA cable though I would recommend HDMI or DVI as text is much sharper and easier to read. Yes you do need to have the 6 pin plugged in or it will not receive enough power to work.
If it boots with only 1 stick of RAM, that stick you're holding may be bad. If it doesn't matter which stick, it may be incompatible RAM. Sometimes they can fail in a way that lets them work intermittently, which may have happened here. Try alternating which stick on RAM is installed. Make sure you're removing power completely (unplugged from the wall) before touching anything in there, and ground yourself to the chassis frequently.
The GPU must have its power connectors all hooked up, or it won't work (as pointed out above).
The chances of two separate components of two different types causing the same type of boot failure are slim when RAM may be the original culprit. If you can get it to successfully boot on integrated graphics with the graphics card removed a few times, that's good news for the RAM. They should be installed into similar color-coded banks, starting with slot 0 and seated firmly.
When the graphics card is reinstalled, make sure it's in the uppermost slot available, properly seated and secured, connected to power, and not bumping the RAM or anything else. When the POST screen comes up, if you have the chance, enter the BIOS and load the optimized default settings.
I have tried one RAM stick in slot 1, the other in slot 1, one it slot 1 and the other in slot 2, and switched (those are the two slots the manual says to use).
All of those configurations without the GPU, work fine.
All of those configurations with the GPU in the PCI slot but without the power connector, work fine.
All of those configurations with the GPU in the PCI slot and with its power connector, cause the computer to not boot.
The only thing causing the computer to not boot is the GPU's power connector. I have even installed its driver from the disk, and it isn't shown in device manager when it is connected, and plugging the monitor into it never works.
Also, the first screen I see when I turn it on says TAB: POST. So I pressed Tab and some screen came up and was gone within a second. I did this with the GPU in the PCI slot but not with the power connected.
I've re-written the OP for clarity, because it was too wordy before.