It Should've Been Lorelai - S2-E14
Continuity mistake: While Lorelai and Christopher are talking at the table at Luke's, a blue coffee mug appears only in one shot but not the rest of the scene.
Dead Uncles and Vegetables - S2-E17
Continuity mistake: When Lorelai and Luke are sitting at his kitchen table, there is a box behind Lorelai with a shirt that goes from hanging out of the box to being in the box repeatedly.
Back in the Saddle Again - S2-E18
Continuity mistake: When Lorelai sits down with Dean to discuss how "clingy" he is with Rory, both Lorelai and Dean's hand positions change between shots, specifically whether their fingers are laced, resting normally on the table or raised in the air.
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)
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.
Lorelai's Graduation Day - S2-E21
Continuity mistake: When Rory and Lorelai are talking in their kitchen, Rory pushes the cafetière plunger down but then over the conversation it is pictured sometimes raised, and then lowered and then raised again a few different times. (00:04:30 - 00:06:00)
Lorelai's Graduation Day - S2-E21
Continuity mistake: When Rory visits Jess she has a cast on her left arm from the car accident, but while Rory's on the bus returning from her trip the cast has vanished, then the cast reappears when Lorelai gets home after graduation. (00:33:15)
Lorelai's Graduation Day - S2-E21
Continuity mistake: When Emily, Lorelai and Richard are going to take the picture, Richard lifts his arm as if he is going to put it around Lorelai's shoulder. The angle changes and his hand is no longer raised. He then raises it again before finally lowering it.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.