Those Are Strings, Pinocchio - S3-E22
Stupidity: Richard, an insurance man drove Rory's Prius graduation gift to Chilton. The car wasn't insured, and presumably wasn't registered. An insurance man who is anal about following procedure drove an unregistered and uninsured vehicle? Richard wouldn't do it. Plus, Richard should have mentioned transferring the title to Rory. Can't insure it until it's in her name.
Always a Godmother, Never a God - S6-E4
Stupidity: Rory arrives in Star's Hollow for the baptism. She parks presumably near the church. When she gets out of the car, she does not have a purse or car keys in hand. She apparently left the car unlocked and left the keys in it. It's just plain dumb.
Stupidity: Lorelai finishes her character reference letter for Luke. She takes it out of the typewriter and immediately folds it. She did not sign it. I wonder why a judge accepted a reference letter that wasn't signed.
Norman Mailer, I'm Pregnant! - S5-E6
Stupidity: Luke arrives at Lorelai's house. He comments that she left her car light on. Lorelai then tells him that she left it on because the porch light is out. He says that the battery is going to die. She tells him that she has it timed out perfectly. She then says that there will be 2 hours to spare and that the car has all day to charge. Cars don't charge unless they are running. Is she really going to run it all day? Why didn't Luke point out the error in her logic? Doing so is staying true to the character. The oversight is glaring.
Chosen answer: Lorelai and Christopher get married impulsively in Paris but they get divorced shortly after returning home. I don't know the exact number of days or weeks but they definitely weren't married for more than a couple of months.