The Road Trip to Harvard - S2-E4
Continuity mistake: When Lorelai returns to Stars Hollow she goes to Luke's, and as she takes a seat at the counter there's a glass domed cake platter (with what looks like pie) at the end of the counter, which promptly vanishes in the following shots. Also, the blue box of wax paper sheets moves to the opposite side of the hot cups on the counter, between shots. (00:37:55)
The Road Trip to Harvard - S2-E4
Continuity mistake: When Luke offers Lorelai coffee, Luke grabs two paper hot cups, but the camera angle shifts and he is holding only one cup, then the angle shifts again and he is back to having two cups while he proceeds to pour the coffee. (00:38:10)
The Road Trip to Harvard - S2-E4
Continuity mistake: When Rory tells Lane about what she saw at Harvard, she comments, "where I might live next year." But earlier, Rory says that she's still two years away from Harvard.
The Road Trip to Harvard - S2-E4
Continuity mistake: Before Lorelei leaves the diner at the end, there is a dirty coffee mug at the end of Luke's counter, but when she leaves, the mug is gone.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.