Other mistake: Lorelai is in bed post breakup. Rory returns from running errands for her. She comes upstairs with Frank who is carrying the television. Frank puts down the television and leaves. When Rory leaves, Lorelei turns on the TV. No one plugged it in or hooked up the cable, assuming that cable is installed in her bedroom (which is extremely doubtful). Lorelai would not have been able to flip through channels as she did.
Continuity mistake: When Rory and Paris sit down to eat cereal in the cafeteria, the table has dirty dishes on it, but when they call over the girls, the table is clear.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.