Continuity mistake: When Chris offered to make coffee, and he goes to the water stand, you can see there is already water in the coffee pot.
Continuity mistake: When Rory and Jess go for the ice cream run the rear-view mirror is missing, but when Rory tells Jess to turn right the rear-view mirror is attached to the windshield in the exterior shot. (00:25:30)
Visible crew/equipment: When Luke brings over the box of brownies for Lorelai he walks in and the front door is closed, then as they both walk away we can see the reflection of the boom pole moving on the door's glass. (00:13:25)
Audio problem: When Rory and Jess are talking in the car, there's a mismatch sound when Rory says "whatever" followed by "wherever". (00:19:00)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.