|The initial problem you describe is almost always either the PSU, motherboard or both (since a bad power supply can ruin the motherboard as well). Your CPU may not have worked in the other computer because it wasn't compatible with the motherboard or not configured properly on the motherboard. Or it might be bad too.|
Also a psu may appear to be good in that when switched on the fans spin up, LEDs light up and drives spin but that's no guarantee its working. As already mentioned it has to provide several different voltages at sufficient current in order to be 'good'.
It's kinda hard to follow the disjointed troubleshooting comments in your # 2