Continuity mistake: When Lorelai and Max are on the porch talking about whether or not Max can go in, there are a few snowflakes on Lorelai's hat which disappear and reappear while they're talking.
Continuity mistake: During the town meeting, when Harry is talking about the Battle of Stars Hollow, in the shots facing Harry his hat is sitting on the top edge of the lectern, but in the shots from over Harry's shoulder the hat is sitting on the lectern's inner shelf. (00:01:15)
Continuity mistake: When Rory comes home from her grandparents and talks to Lorelai about Max sleeping on the couch, into the kitchen and the red-white step stool chair is beside Rory's bedroom door, but when she and Lorelai walk into the kitchen it's gone, then it reappears after Lane leaves. (00:39:45)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.