Continuity mistake: When Detective Shale knocks at Fred's door, he is wearing a blue suit, but when he enters the door, his suit is now yellow.
Continuity mistake: When Fred and Wilma are in their beds, you can see that the holes in the dial of the phone are in a diagonal position, but when we see a close-up of the phone, the holes are now straight.
Answer: The name of the pet is "Baby Puss." He has appeared in episodes, but only a very few. The first episode was probably "The X-Ray Story" from season 2.
Bishop73