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)
Visible crew/equipment: When Lorelai and Rory are trying to get to Luke's they wind up in the alley, and in the next shot inside Luke's we can see the reflection of the moving camera on the glass door, behind the guy wearing a blue check flannel vest. (00:11:50)
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)
Visible crew/equipment: When Lorelai and Rory are in the kitchen talking about the to-do list, the camera is visible in the reflection on the microwave. You can see the cameraman's hand rotating the control to move the camera as Lorelai sits down. (00:07:20)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.