What you can do is, you should test everything outside of the case first, before screwing anything into the case. You really only need to plug the CPU, RAM, CPU fan, video card, and power supply to the motherboard in order to test for power and screen (
see pic here). Make sure all power cables are plugged into the video card and motherboard (there should be at least 2, one is the big 24 pin one, and the other should be either a 4 pin or 8 pin near the CPU). Then you might ask, how do you turn the system on? Very simple, I supposed you have already found the
2 pins for power switch on your motherboard. To emulate a power switch, all you need to do is get a flat head screw driver and short those 2 pins like
this picture. It might look a bit risky and shady at first, but you'll soon find out that when the 2 pins are shorted, the system will turn on. Of course, make sure your power supply is
switched to on position.
You can plug your keyboard into the USB port on the motherboard if you see a boot screen. Turn off the entire test setup by switching the power supply to off.
I can't be sure if the 3-prong to 2-prong power adapter is affecting your system. I don't have one handy to test it out myself, but I would really try to find a 3-prong outlet and test the computer on that.