Those Lazy-Hazy-Crazy Days - S3-E1
Continuity mistake: In the first shot you can see Rory's laptop open in the background, but when she walks over to her desk and sits down, it's closed. (00:02:15 - 00:04:45)
Those Lazy-Hazy-Crazy Days - S3-E1
Visible crew/equipment: When Lorelai and Rory are on their way to the festival they stop at Sookie and Jackson's house, and as they approach the front door, the reflections of the camera operator and camera are visible on the door's window. (00:25:05)
Those Lazy-Hazy-Crazy Days - S3-E1
Continuity mistake: After Luke and "Mimi" talk at the diner Lorelai comes home, and while she and Rory sit on the couch note the open window behind them, with a clown pillow in front of it and the porch chair on the other side, but when Lorelai asks for her purse because she's hungry it cuts to the exterior shot, and nothing is blocking the view through that window. (00:41:00 - 00:43:00)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.