Continuity mistake: When Lorelai and Luke are in Luke's Diner to choose the color for the walls, Luke turns around the 'closed'-sign on the door. In this shot the venetian blind at the door is quite messed-up; partly closed, partly opened. In the following shot, where all the people are standing outside Luke's Diner looking through the windows, the venetian blind is orderly and completely opened.
Continuity mistake: In the first scene when Dean brings pizza and they are discussing about Donna Reed, in one shot Lorelai is opening the pizza box and she and Rory take a piece and start eating while they are talking. In the following shot, the piece of pizza in Lorelai's hand is gone and Rory takes a piece from the box - you can see that the pizza is complete again.
Continuity mistake: Dean takes out his salad box when they sit and watch the Donna Reed show, but he doesn't open it. In the next shot it is open, and in the next shot again, a salad leaf is at his fork, without touching it.
Continuity mistake: While Kirk asks Luke if he is really painting the diner, Luke adds napkins and closes the to-go box twice.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.