Plot hole: When Emily is dialing a number for Manny Martelli, to make arrangements for the funeral, she only dials two digits. She just got the number from someone else, so there's no way it was speed-dial.
It's Just Like Riding a Bike - S7-E19
Plot hole: The dog, Paul Anka, uses the porch steps without a blanket over them. In an earlier episode, Lorelai states that he's afraid of the porch steps when Babette tries to walk him. (00:32:50)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.