Continuity mistake: At Luke's, when Rachel shows Lorelai the photo of the Dragonfly Inn, Rachel says, "It's an old abandoned inn I ran across when I was exploring the other day." So Rachel took the photograph in the last few days, and in the photo the Dragonfly does not have shutters, and does not have a sign hanging over the porch, etc. When Lorelai visits the Dragonfly with Rachel and then with Rory, the shutters and sign have appeared, as well as other changes. (00:04:50 - 00:40:20)
Continuity mistake: Lorelai and Luke are talking in the storage room. The turkey gravy behind Luke keeps rearranging itself.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.