Revealing mistake: While Rory and Lorelai are in the kitchen, in a few closeups of Rory we see the refrigerator and counter area with the microwave, etc. behind Rory, but when Lorelai is at the back door about to join Luke, part of the set beside the refrigerator is gone. (00:04:15)
Those Are Strings, Pinocchio - S3-E22
Revealing mistake: Rory and Lorelai stop looking for a place to carve their initials on the Chilton steps because they hear someone coming. A man walks by and his footsteps stop so they continue, but when the screen goes back to Lorelai you can see the man hiding in the plant behind her on the right.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.