Car electrical systems are often funkier and harder to diagnose than home electrical. A bad ground somewhere can cause voltage to back-feed through other systems to get the necessary electricity.
Some of the oddities you describe, such as unrelated parts being activated for no reason (turning on headlights/wipers causing doors to lock, for example), are almost always caused by a bad ground somewhere. First thing I would do is make sure any wires that bolt directly to the chassis are clean and tight.
Ever seen a car (usually VW, Volvo, Saab) with a turn signal on, and the tail light and backup light on the same side flash dimly? Classic ground problem, as the voltage goes through the other bulbs to complete the circuit.
My mother'* '97 Cavalier had no driver side low beam, high beam and DRLs were fine, and a new bulb and socket didn't cure the problem. I had 12v coming to the socket when low beams were on, but as soon as the bulb was connected, the voltage went to 0. I found a corroded and frayed wire going from the driver side to the passenger side low beam, repaired the bad section, and voila! Perfectly functioning headlights.
Good luck, I hope this helps you fix your Cavvie!