Other mistake: When Christopher shows up, he says "tomorrow is Saturday." Wouldn't the girls be at Friday night dinner? Not ordering pizza?
Visible crew/equipment: When Lorelai argues with Richard in his study, nothing gets resolved and Richard says that he's tired, then in the last shot of Lorelai we can see a crew member's fingers holding equipment, at the right side of the screen. (00:25:55)
Visible crew/equipment: When Luke is waiting for Lorelai to show up and paint, a crew member, camera, and boom mic are reflected in the window next to him. (00:29:45)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.