Like Mother, Like Daughter - S2-E7
Visible crew/equipment: When the guidance counselor introduces herself to Rory during lunch she tells Rory to meet with her after lunch, and when she walks away camera equipment is visible at the bottom left corner of the screen. Then, at the end of the episode, when Lisa asks Rory if she can join her for lunch, the camera equipment is visible again before it moves. (00:06:00 - 00:43:15)
Like Mother, Like Daughter - S2-E7
Visible crew/equipment: After the Mrs. Backward baseball cap line Lorelai goes to leave. Luke says "and by the way" and there is a camera part showing in the top left corner. (00:42:15)
Like Mother, Like Daughter - S2-E7
Continuity mistake: When Mr. Charleston is looking at Lorelei's parent file, he closes it and starts talking and in the next shot he is seen closing it again.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.