yeah, pairing up two of those 4 pin connectors to make an 8 pin is what you do with some PSUs. making sure of one thing though, your old PC wasn't a dell, right?
seems that there can be three problems. 1 - psu 2 - motherboard 3 - cpu. PSU you said worked before, which checks it out, unless it was a Dell PSU. Motherboard, eh, see if it has a display on it. Should be a small square that can display two numbers at once, sorta like how a digital clock displays numbers. If it has one, which it should, watch it when you try to start the computer up and see if it does anything, gets stuck at a number/code, etc. The CPU I'm suspecting as the most likely culprit. Get the cooler off, remove the CPU, and make sure nothing's bent in the socket, nothing looks wrong, and make sure it goes in right. Reapply TIM, don't forget. Never reuse old TIM.
Last thing is that you mentioned slow spinning fans? There's a slight possibility that you may have overloaded the capacity of the motherboard's fan power load and err, fried it. No high-power fans or anything, right? How many fans? Are the fans connected to the power supply directly, or on the motherboard? If on the motherboard, remove them all, and try to start it up. If on the power supply, again, remove them all, and try to start it up.