Continuity mistake: When Lorelai and Luke are talking in bed she is wearing a robe. It goes from on to off her shoulder with each shot.
Continuity mistake: When Lorelai enters the kitchen and sees Rory cooking the juicer is visible, but in the next shot you see Rory go to the drawer and get it. (00:25:24)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.