Visible crew/equipment: When Lorelai is in the diner questioning Luke about Nicole, during their conversation the reflection of the boom operator holding both arms up is visible on the window. (00:21:40)
Visible crew/equipment: When Lorelai is doing her baby-talk with "wittle Michel" he continues walking as she stops in front of Luke's, and the reflections of the camera operator and camera are visible on the glass door. (00:16:10)
Continuity mistake: In the scene where Rory is talking to Lane on her cell phone, she sits down on a bench. On that bench is a guy reading a book. The scene shifts to Lane in Stars Hollow, and when it shifts back to Rory at Yale, the guy on the bench is gone. (00:16:40)
Continuity mistake: When Rory is sitting in the theatre, on the phone to Lorelai, we see the same people walk behind her twice - the woman with white jeans on and the woman with beige trousers. (00:42:10)
Continuity mistake: While Lorelai is at the diner talking to Luke about Nicole, when she takes a seat at the counter she places her black purse on the stool beside her, but in the following shots that purse repeatedly reappears and disappears from her shoulder, depending on the camera angle. (00:23:25)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.