Dear Emily and Richard - S3-E13
Factual error: When Lorelai arrives at the hospital, Rory says she is xeroxing a report that she needs to fax for Sherry because Sherry didn't make enough copies. No matter how many people the report needed to be faxed to, only one original copy would be needed.
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.