Dead Uncles and Vegetables - S2-E17
Continuity mistake: When Lorelai and Luke are sitting at his kitchen table, there is a box behind Lorelai with a shirt that goes from hanging out of the box to being in the box repeatedly.
Sadie, Sadie - S2-E1
Continuity mistake: When Lorelai and Rory get home from the upsetting dinner from Richard attacking Dean, while sitting at the table one shot shows the Twinkies arranged on the plate in front of Rory on the table. In a few other shots, you see a few twinkies missing and not arranged.
Presenting Lorelai Gilmore - S2-E6
Continuity mistake: When Christopher comes to the house and they are all outside talking, there is no garage on the side of the house. Only gates. In later episodes there is a garage.
It Should've Been Lorelai - S2-E14
Continuity mistake: While Lorelai and Christopher are talking at the table at Luke's, a blue coffee mug appears only in one shot but not the rest of the scene.
Continuity mistake: When Lorelai puts her book bag on the table there's an economics book showing and in the next shot it is gone.
Nick & Nora/Sid & Nancy - S2-E5
Continuity mistake: When Rory interviews Max, in each shot the pens on his desk in the cup keep rearranging.
The Bracebridge Dinner - S2-E10
Continuity mistake: When Rory is checking in Lane and Mrs Kim, twice in a row it shows her close her folder and hold it at her side and in the next shot it is still open.
Like Mother, Like Daughter - S2-E7
Continuity mistake: When Mr. Charleston is looking at Lorelei's parent file, he closes it and starts talking and in the next shot he is seen closing it again.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.