My first thought would be that the power supply is bad. If that isn't it, then the motherboard is bad. Something is telling the 12v rails to turn on in the power supply all the time, regardless of whether the motherboard is powered up or not, and so all 12 volt items are running wide open. Once the motherboard is powered on, it dials the RPMs on the fans down based on the temp of the sensors on the motherboard and the settings in the bios. I have seen this a few times, and it is usually one or the other.
An ATX power supply needs a signal from the motherboard to turn on, but the 12v rails in yours are powering up without that signal. So to troubleshoot, I would open the case and unplug the main 24 pin harness from the motherboard but leave all other things plugged in. Then plug the power supply in and see if the fans come on. If they do, it is the power supply. If they don't, it is more likely to be the motherboard that is signaling the power supply to come on but not actually turning itself on, and the motherboard is the likely culprit.
I would guess with about 95% certainty that your computer uses a standard ATX power supply, but there is a very small chance that they used something proprietary, usually when that happens it is a standard PSU with a few pins different on the motherboard connector. You should be able to track this information down pretty easily though.
As for mame, I would think that a gen1 core i3 cpu would be more than sufficient to run any version of mame. There might be a couple games that won't run well, but overall I wouldn't hesitate to run the latest version, I have run .153 and .154 on half the CPU of an i3 with no difficulties.