Visible crew/equipment: After dinner with Emily, when it cuts to Lorelai and Rory rushing into their house, an equipment pole is visible at the right side of the screen at the start of the shot, and a standing spotlight lamp is reflected in the wall mirror. Then as Lorelai walks back down the stairs the camera is visible at the left side of the screen, and when Lorelai exits, the spotlight lamp's reflection is visible again. (00:06:25)
Visible crew/equipment: When Rory (holding the muffins) and Lane are walking on the porch heading to the garage, the reflections of crew, equipment, and the camera operator are visible on the windows. (00:11:05)
Character mistake: In this episode, Rory tells Lorelai that she was cleaning her computer's hard drive because it was full of unopened downloaded files which were consuming her CPU speed. This is wrong, the situation which she's explaining would not use up any extra CPU speed - she may be trying to say her hard drive was extremely fragmented which MIGHT consume some CPU speed (but a tiny amount).
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.