When Miss Patty sees Doula in the diner she gushes about the baby's beautiful eyes. In the next shot we see that the baby is asleep, with her eyes closed.
Luke's lawyer states that there is no way the court will consider upending the current state of April's custody to be in her best interest, and that it doesn't matter that the current arrangement exists only because Luke didn't know about her.
In fact, it is down in Connecticut law that a court should be prejudiced in favor of an existing custody arrangement that has been determined BY PREVIOUS COURT DECISION OR PREVIOUS AGREEMENT. This is not how Anna got full custody of April, and therefore the fact that Luke didn't know that he even had a daughter does indeed matter in the eyes of the law. A real lawyer would have known this.
In fact, it is down in Connecticut law that a court should be prejudiced in favor of an existing custody arrangement that has been determined BY PREVIOUS COURT DECISION OR PREVIOUS AGREEMENT. This is not how Anna got full custody of April, and therefore the fact that Luke didn't know that he even had a daughter does indeed matter in the eyes of the law. A real lawyer would have known this.
At 13:30 Lorelai is holding Liz's newborn Doula, when after a camera angle change she is clearly now holding a doll. This doll's legs and torso are nearly flat, clearly seen as Lorelai pulls it close to herself.
For someone as smart as Rory, who attended Yale and reads a lot of books, they made her not know of "Men Are From Mars, Women Are From Venus" when she should be smart enough to know that book.