Continuity mistake: When Barney is at the pay phone, there is a dial in the centre of it, but when he sees the car going off on its own with Fred in the driving seat, we cut back to Barney and the dial has disappeared.
Continuity mistake: When Fred picks up the receiver on the telephone, there are five holes on the dial, but when we a close-up of the dial in the next shot, there are now six holes on it.
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