Continuity mistake: While Lorelai and Rory are at Luke's for a very early breakfast, when Miss Patty walks in she's wearing long earrings, but the earrings vanish and then reappear while she's telling Lorelai how badly she feels about Rory and Dean. (00:13:15)
Continuity mistake: When Babette walks through the kitchen's back door to comfort Rory, the porch is visible with its lattice wall and doorknob is to the right, but when Lorelai walks out with Babette in the exterior shot the wall has changed and the doorknob is to the left. The exterior shot looks like the area outside the side living room door, not the kitchen door. (00:19:00)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.