Ten Little Flintstones - S4-E16
Continuity mistake: When the spaceship lands for the second and last time during the episode's finale, it appears to be night-time, but when Fred is chasing all of his duplicates down the street, it appears to be daylight.
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