Fight Face - S6-E2
Other mistake: While in Luke's diner, Luke hands Lorelai a cup of coffee in a 'to go' cup. When Luke puts the coffee in front of Lorelai, the cup makes a hollow sound when it hits the counter, obviously indicating that there's no actual coffee in the cup.
Twenty-One Is the Loneliest Number - S6-E7
Other mistake: Though I love the 21st birthday at the casino plan, it's flawed. Rory will not be able to play blackjack until she is 21. She can't be playing when she turns 21. Casinos, to the best of my knowledge, won't let you on the gaming floor unless you are 21. She has to turn 21 first.
Twenty-One Is the Loneliest Number - S6-E7
Other mistake: After Emily goes to the pool house on the pretence of looking for a blouse, she asks Logan about plans for Rory's birthday. Later, Emily and Richard talk. Emily suspects that Rory is ready to have relations with Logan. She says that she caught them getting cozy on the couch. Emily didn't catch them doing anything. There was no way for Emily to see inside. Both Logan and Rory got off the sofa to answer the door. Emily didn't see anything. And, Logan told Emily that he had no plans for where his birthday. He knows that Emily is planning a party, yet he offered to take Rory away for her birthday. Emily would've been devastated.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.