My UPS doesn't seem to be handling the power outages very well.

The UPS may have a bad battery. Follow the instructions per the manufactuer to fully charge the battery. The unit could also be overloaded. Try taking some of the hardware off and see if that helps. You may also want to try a different wall outlet. All other things checked, though, it is probably a bad UPS.

The power supply fan isn't working or works intermittently or is noisy.

You probably need to replace the power supply, but first, clean the dust out of the supply using a vacuum and/or compressed air and see if it goes away.

The power supply is whining.

It is probably your CPU fan, in which case you can troubleshoot as above. If it isn't the fan, then there is a problem with some electrical component within the power supply. This would warrant replacing it.

Sometimes the electricity in my house dims and my PC reboots.

The PC gets a power brown out and it thinks the power is turned off, so it shuts down. Then, when the power comes back, even a second later, it thinks you turned it back on, so it tries to reboot. The solution is to get yourself a backup power supply, or UPS.

When I turn the PC on, nothing happens. The power supply does not turn on.

Well first, make sure the unit is plugged in and actually getting power. If you have an ATX system, you need to check the case connectors from the power switch to the motherboard's Power Switch connectors. Very often, especially when just building an ATX system, these connectors will be incorrectly connected and will cause the PC to just sit there dead. You could also have a short circuit in the case, so check for this as well. If you rule these things out, then the power supply could be bad.

I want to connect another device to my system but am out of power leads.

Wow, you must have a very full system. But, this is no problem. Head down to your local PC Madness Store and pick up a Y-connector. You can use this to turn one power lead into two.