Women of Questionable Morals - S5-E11
Other mistake: When Rory goes to see her father after Strobe dies, she brings a paper bag with cookies. Christopher comments "and milk", even though the bag was creased. He couldn't see into the bag because the bag is closed.
Other mistake: Lorelei, Rory, and Luke are about to leave for Emily and Richard's wedding. Babette is on the sofa half passed out from the bachelorette party. When she hears Lorelei speak her name her head pops up to say goodbye, and she also comments that Luke looks nice. Luke is wearing a long overcoat. She could not possibly see how he looks, plus she was viewing him from behind.
Other mistake: Lorelai is in bed post breakup. Rory returns from running errands for her. She comes upstairs with Frank who is carrying the television. Frank puts down the television and leaves. When Rory leaves, Lorelei turns on the TV. No one plugged it in or hooked up the cable, assuming that cable is installed in her bedroom (which is extremely doubtful). Lorelai would not have been able to flip through channels as she did.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.