Jeff, you're awesome. Whatever went wrong burned out about two years ago. I also keep the shutter closed and keep most of the games in their plastic sleeves. I'll look into the eBay bit. That's great. I have hope for it now.
The 72-pin connector is not an electronic part, but just connects the cartridge to the motherboard. If the system won't power on (with or without a game present), the connector isn't the cause. Be sure to check with a different AC adaptor, in case it's that. If the control deck is bad, then I don't know whether it's worth attempting to repair it; in that case, a replacement might be easier.
As for games, they can be easily cleaned with cotton tips and alcohol. Use 70% concentration. If using 90%, dilute it with distilled water. (Nintendo recommends diluting even the 70% concentration. They have said that there is a remote possibility that alcohol can crack the plastic, so to minimize that chance, use another cotton tip to dry the connectors and plastic afterward.) Blowing on the connectors can slowly corrode them. To clean the console connectors themselves, you'll need a cleaning kit, which has a rigid cloth insert (a kit exists for each console). Third-party cleaning kits exist; use caution with them.
