Those Are Strings, Pinocchio - S3-E22
Continuity mistake: During the graduation Lorelai borrows Richard's pen and writes "we got the inn" on the graduation program, and when she holds the program up to show Rory note the position of the blue logo printed on the bottom page. First the logo is near the center crease, but then it's at the outer edge of the page. (00:35:20)
Those Are Strings, Pinocchio - S3-E22
Visible crew/equipment: When Lorelai, Sookie, and Jackson are "celebrating" they all sit down at the table, and we can see the T-mark on the floor behind Jackson. (00:11:30)
Those Are Strings, Pinocchio - S3-E22
Revealing mistake: Rory and Lorelai stop looking for a place to carve their initials on the Chilton steps because they hear someone coming. A man walks by and his footsteps stop so they continue, but when the screen goes back to Lorelai you can see the man hiding in the plant behind her on the right.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.