Continuity mistake: In the scene where Luke and Jess are on the lake discussing in the end, when the camera is on Luke they are in the sun and when it is on Jess they are in the shadow.
Visible crew/equipment: When Lorelai returns from her trip she tells Rory that she needs food, and as they walk away from the car, we can see the reflections of the camera operator, camera, and boom mic moving on the surface of the car. (00:38:30)
Visible crew/equipment: While Dave, Brian, and Zack are in the car, when Mrs. Kim is speaking to Dave the reflection of the boom pole is visible in the window behind Mrs. Kim, then after she leaves when the car starts moving, the reflection of the reflector screen is also visible in Brian's window. (00:31:15)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.