2002 Rendezvous with haywire electrical system!

I own a 2002 Buick Rendezvous CXL, with apx. 60,000 miles on it. Since I bought it in March 2005 I have noticed, when driving at night, the lights dim - sometimes almost to the point of going out - then come back. What I now find most troubling is that when I drive for more than 1/2 hour, my electrical system goes haywire. It tells me that my e-brake is on (it isn't) and the alarm won't stop sounding. It also registrers that a door AND the rear hatch are ajar (they aren't), the fog lights start flashing, the auto locks keep trying to re-lock, and my dome lights come on and off. The last time this happened, the car would not stop, not even after turning it off. The headllights continued to flash and the alarm inside the car would not stop. We had to disconnect the battery to stop the commotion. However, each time it happens - once the car cools down it seems to stop. If I'm only taking short trips within the city, everything is fine. Do you have any idea what the problem could be?