Mac Pro (2006-early 2008 model) will not boot

Hey everyone! So, I recently got this Mac Pro (Model number A1186, EMC 2113) with two Quad Core 2.66 GHz Intel Xeon 5150 processors, 16GB of RAM (both RAM riser slots completely filled with 4x2GB sticks of 667MHz DDR2 ECC FB-DIMM each), 2x1TB clean HDDs, and an ATI Radeon HD4870 (512MB DDR5 memory) video card with 2 DVI ports and an S-video port. (I bought the HD4870 later after I found that the one that came with it, an ATI X1900 XT, was part of a recall [sadly this recall ended in 2011.] Plus, the fans on the X1900 were revving up REALLY loud.)

The reason I'm posting here is because the darn thing will not "boot." The machine starts up fine (HDDs spin up, lights on the RAM risers illuminate momentarily, etc.), but there is no output out of either of the DVI ports. Not even a gray Apple screen. I have tried the DVI cord and the monitor on two different machines and they both work perfectly fine, and now I have no other Mac available to test the video card on. I'll just have to assume for now that it works. The video card, after the machine is turned on, has three of the four diagnostic LEDs lighted. The ones that light up are D1601, D1602, and D1603; the other one, D603, does not light up. After some searching around on Google for a little while and sifting through forum after forum, they all seemed to agree upon the fact that these lights were from the two internal power cables (D1602 and D1603) not being plugged in and that there was a critical temperature error (D1601). Note that a) both power cables are in both power slots and hooked in securely and b) these lights turn on immediately after pressing the power button. A light tug from both ends on both cables assures that these cables are not going anywhere soon, and I doubt that there would be such a heating issue as soon as I press the power button, especially since all the fans are running fine. My first thought was that maybe the logic board wasn't getting enough power, but I have no idea on how to check this as almost everything besides the RAM risers and the extension slots are covered by something.

Another problem I found is that some of the diagnostic LEDs that should be lighting up, aren't. Out of all the lights, the ones that do light up are the TRICKLE light and the EFI light. I guess it's good that the CPU lights aren't lighting up, but what about the the GPU and POWER lights? Shouldn't those be lighting up too?

I also tried booting up from a Snow Leopard disc (there isn't any internal CD/DVD drive [the previous owner removed it for whatever reason], so I've been using an external CD/DVD drive). When I plugged in my Apple USB keyboard, I decided to press the caps lock key and see if it would light up green. No dice. I tried all 5 USB ports on the machine and none of them worked. But how did the CD/DVD drive work? It spun up when I turned the machine on. So since the keyboard can't be recognized I am unable to boot from a CD, or anything else for that matter.

One more issue: there is no startup chime! Tried turning it off and on many different times trying some potential solutions out and not once was there a startup chime. I know for sure this is a huge issue.

With all of these huge issues, I tried a few of the recommended solutions: reconfiguring the RAM (which I inserted per Apple's instructions), resetting the SMC, resetting the PRAM (which wouldn't have worked anyways since the keyboard isn't recognized), and resetting the NVRAM (which, again, probably wouldn't have worked since the keyboard isn't recognized.)

So here are my problems. I've asked Mac Rumors, r/applehelp on Reddit, and sifted through search after search on Google and I can't find the solution. Anybody know how to fix this machine? Any help is greatly appreciated.