Visible crew/equipment: While Lorelai and Luke are in the diner's storeroom, just as Lorelai makes a comment about not knowing what's going on in Rachel's head because she's not a Vulcan, the boom mic dips into view over the white container on the top shelf, at the top right side of the screen. (00:30:00)
Other mistake: Rory asks Emily for baby pictures of Lorelai, but Emily says Lorelai burned them all because she had such a big head when she was little. But in "Love and War and Snow" earlier in the season, Emily, Richard and Rory looked at a baby picture of Lorelai in a bunch of petticoats the night Rory stayed over with her grandparents during a snowstorm.
Visible crew/equipment: When Lorelai talks to Emily at the Independence Inn, a boom mic briefly drops into the shot. (00:21:25)
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)
Other mistake: While Luke and Lorelai are talking about Rachel in the storage room, after Lorelai places some jars on the shelf, the jars on the shelf behind Lorelai turn, change position or vanish, between shots.
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.