Visible crew/equipment: After Rory tells Lorelai that she looks happy, when it cuts to Max telling his class about Emily Dickinson's poem, we can see the boom mic bobbing around at the top of the screen. (00:07:55)
Visible crew/equipment: While Lorelai is cleaning out the fridge, when she picks up the book and asks Rory to give the book back to Max, lighting equipment is visible at the right side of the screen.
Visible crew/equipment: A mic drops into shot just as Jackson accepts Sookie asking him on a date. (00:38:45)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.