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.
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.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.