Continuity mistake: In the episode when Jess arrives at Luke's, Luke explains to Lorelai that Jess' father walked out on his family two years ago. Jess is 17 at that point, which would make him 15 when his father left. In the episode where the father comes to see him, Luke asks him why he left when Jess was just a baby in diapers, and indicates that he hasn't seen him since.
Continuity mistake: This was in season six in the Martha's vineyard episode (Valentine's day). Lorelai comes into the kitchen and we see a shot of Rory cooking. Next to her is the orange juicer. In the next shot, it is not there. It reappears as Lorelai is watching Rory cook and Rory reaches into a drawer and pulls it out.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.