Visible crew/equipment: After Sookie dumps her pan of ratatouille in the sink she suggests going to Cicero's, and just as Jackson says, "They have good ratatouille. OK, so long as I can keep my shoes on," the reflection of the dolly grip pulling the camera dolly is visible on the window. (00:38:20)
Continuity mistake: After Lorelai and Christopher bring coffee and donuts to Rory and the other newspaper staff, while Rory's talking to her parents she's holding a blue pencil and papers with no clipboard in sight, but when Chris and Lorelai say goodbye Rory's pencil is black and a clipboard has suddenly appeared in her closeup. (00:40:50)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.