Fight Face - S6-E2
Other mistake: While in Luke's diner, Luke hands Lorelai a cup of coffee in a 'to go' cup. When Luke puts the coffee in front of Lorelai, the cup makes a hollow sound when it hits the counter, obviously indicating that there's no actual coffee in the cup.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.