Continuity mistake: Lorelai and Luke are having a romantic dinner in his apartment. She puts candles in the sticks at an angle, but when they return to the table the candles are straight in their holders. (00:18:00)
Visible crew/equipment: While Liz and TJ are arguing in Luke's bathroom, when Luke puts the lid on the plastic container with the cheesecake for Lorelai to take home, we can see the shadow of the bouncing boom mic on the cabinet over the stove. (00:36:25)
Visible crew/equipment: When Rory comes out during the party to meet Dean, he says, "I don't belong here, not anymore, do I?" then as he says, "You look good," a boom mic drops over his head. (00:39:20)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.