Visible crew/equipment: Near the end of the episode when Rory comes home and is standing on the porch talking to Lorelai at the door, right before Rory says "Hi" you can see a crew member in the background, standing in the bushes on the other side of Lorelai's jeep and then moving out of sight (toward the left side of the screen). (00:39:20)
Visible crew/equipment: A mic drops into shot just as Jackson accepts Sookie asking him on a date. (00:38:45)
Rory's Birthday Parties - S1-E6
Visible crew/equipment: As Lorelei climbs into bed with Rory at 4:03 there is what appears to an open book being lowered out of the shot by a presumed crew member.
Visible crew/equipment: When Rory is dropped off in the morning, he walks up the front porch, Lorelai greets her at the door. Behind Rory across the driveway in the bushes is a man. The many slowly hides back into the bushes. (00:39:19)
The Lorelais' First Day at Chilton - S1-E2
Visible crew/equipment: Right before Miss Patty asks Lorelai if that was what she wore at the Chilton, there is a boom mic's pole showing at the top right corner while she is speaking to the dance class. (00:17:10)
Star-Crossed Lovers and Other Strangers - S1-E16
Visible crew/equipment: While Patty and Taylor argue about the fire and Stars Hollow, there's a mic dip over Lorelai's head right before Luke says "she's bi-polar. And again when Patty says "she is?" (00:13:25)
Answer: Yes, Lane lives with both her parents. Her father does live with them, but he was only seen in the Netflix revival.