Some kind of cooling problem or logic board?

It sounds like the capacitors on your iMac are failing. These model iMacs are known for having the issues that you are describing. I recommend opening up the iMac and checking the capacitors, you may find that there are capacitors that are either beginning to expand at the top of them or that they have exploded. There are kits that are sold online to replace the capacitors but given the age and work involved, I don't think its a good idea, unless of course you have the time and want to take on a project! Search around for "iMac G5 capacitor replacement" for more info on that. Hope this helps!