Continuity mistake: Kirk introduces himself as "Mick" in the second episode. From episode 3 and throughout the series his name is "Kirk."
Continuity mistake: After Rory is walking with Dean, it goes to a shot of Rory and Lorelai eating at Luke's Diner from outside. Rory is wearing a light blue sweater and Lorelai has a black shirt on. Then it goes inside the diner and Rory is wearing a dark bluish/grey sweater and Lorelai is wearing a purple shirt. The first exterior shot is used from the very last scene of the episode. (00:27:05)
Continuity mistake: The appearance of the Gilmore mansion is very different in this episode than in following episodes. Here there are steps leading up to the front door, but none in later episodes, and the area people park their cars changes significantly as well.
Continuity mistake: In the opening scene Lorelai crosses the street to Luke's which is in the middle of the block instead of on the corner of the block.
Continuity mistake: In the shot where Lorelai and Rory are in Luke's diner, after Lorelai moves her burger in front of her, the plate changes positions several times between shots.
Continuity mistake: In the opening scene, and all episode, Luke's diner is considerably smaller than the rest of the series. He has a shelf with random hardware and fishing items that is a window the remainder of the series.
Continuity mistake: Lorelai walks into the kitchen and tells Sookie "she did it" and only has a brown bag in her hand. In the next shot she has a brown bag in one hand and the Chilton letter in the other.
Continuity mistake: There is no building next to Luke's in the pilot, but in subsequent episodes there is one, which he buys.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.