Continuity mistake: While Lorelai and Sookie are with Emily doing the test meal, when Emily tells Lorelai to remove the plates she picks up the two plates and walks out of the dining room, but in the next shot the plates have switched hands. (00:23:35)
Continuity mistake: Lorelai and Sookie take some containers out of the car and head towards Emily and Richard's door. When you see the next shot of them at the door, the containers Sookie is carrying are different from the ones she took out of the car, without her rearranging them.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.