Kirk, working at Doose's, chides Miss Patty for sampling from the produce section, but clearly is not familiar with who she is. In later seasons, multiple episodes make clear that Miss Patty has known Kirk since he was a child; for example, in one episode late in the series, Kirk performs a modern dance routine at one of Miss Patty's recitals and it is mentioned that he created and performed the routine 25 years before at Miss Patty's studio.
In Kill Me Now (2000), Babette refers to Cinnamon as a male cat. In this episode she says that Cinnamon is female.
When Lorelai and Rory are talking about dating Max in Babbette's house, Rory is standing up, then sits down and then, when she stands up again, you can clearly see that her back hair is constantly changing. First, she has a lot on her back, then a little, and so on.
While Max and Lorelai are talking in the coffee shop, the shot from behind Max shows his left hand on Lorelai's arm. The shot from behind Lorelai shows Max gesturing with his left hand. This is repeated several times.
At the 12:36 mark, when Rory and Lane encounter Babette bringing Cinnamon home quickly after their bad experience at Al's Pancake House, there is a woman pushing a stroller behind Rory and Lane. You can clearly see it is a toy baby and not a real baby.
Around the 12:35 mark when Rory and Lane are walking and talking near the grocery store, a lady walks by with a baby in a stroller but the baby is very visably a doll.
The vet mentions that Cinnamon was 260 in human years. First of all, she should have said in cat years. Second of all, 260 in cat years is roughly 50 human years. No cat can live 50 years. (Although, you might argue that Cinnamon was a very special cat.)