[OttoSchultz]

Whenever you have weird electrical gremlins start with the obvious items: check that the battery terminals are clean & tight, then check that the ground under the battery tray is clean & tight. You could try replacing the Relay 109 and see what happens, it's a $10 part from idparts.com. http://www.idparts.com/catalog/product_info.php?products_id=350

Move on to the battery itself and the alternator. How old is the battery? If all of the connections are good, put a voltmeter across the battery terminals with the engine OFF and again with the engine RUNNING. You should have 12VDC with the engine off and somewhere around 14VDC with the engine running. If you have less than 12VDC with the car off, the battery may be bad. If you have less than 14VDC with the engine running, the alternator is probably bad. Any competent mechanic should be able to load test the electrical system.

 

[OttoSchultz]

There's no way to test the relay that I know off. Relay 109 controls the main power supply to the ECU, so when it gets flakey, all sorts of weird things happen. Since it's so inexpensive, most people just replace it. If you've never changed it in the past, it might be worth the $10.

For the main ground just follow the negative cable to where it attaches to the chassis. You may need to remove the battery and tray to get really good access to clean it up.

 

[OttoSchultz]

Is there any screeching? It sounds like an alternator/regulator problem. The alternator pulley is a clutched pulley. It could be getting ready to fail.

Do you have a voltmeter to test the system volatge with while off/running?