Visible crew/equipment: At the start, Lorelai brings home the bags of assorted tacos, and when Lorelai stands at the counter pouring coffee, we can see the red T-mark taped to the floor where she's standing. (00:00:50)
Visible crew/equipment: After Luke admonishes Sookie and takes the ladle away from her, when it cuts to Logan's dining room we can see the crew's mat with blue gaffer tape at the bottom left corner of the screen. (00:26:25)
Visible crew/equipment: In Logan's apartment, when Lanny is telling Logan that Cassandra called, an orange T mark is visible on the carpet to the right of Rory. (00:05:01)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.