A House Is Not a Home - S5-E22
Visible crew/equipment: When Mrs. Kim calls Gill over to look at his van, he tells her it's not big enough for a tour van, and two tape marks are visible on the ground - one where Lane stands and one where Mrs. Kim stands. (00:37:05)
A House Is Not a Home - S5-E22
Continuity mistake: When Lorelai is talking to Richard and Emily about Rory, Richard writes down the name of an attorney and closes his memo pad but in the next shot it is open again with his pen poised.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.