Little Ricky's dog, 'Fred', was played by a Cairn terrier named Danny. Danny was trained by Bob Blair and owned by Frank Inn. Interestingly, Danny also played Mr. Wilson's dog 'Fremont' on "Dennis the Menace".
Fred says that there's a no-pets clause in the lease; however, in earlier episodes, his tenant Mrs. Trumble says that she has a cat.
Lucy catalogs Little Ricky's pets for Ethel as she is feeding them. Their names apparently are mostly show-biz "inside" references. The birds are Alice and Phil (Alice Faye and her husband Phil Harris, singers). The turtles are Tommy and Jimmy (Tommy Dorsey and his brother, Jimmy Dorsey, the bandleaders). (The names for the fish - Mildred and Charles - are not as easy to identify; perhaps they were friends or family members of Lucy or Ricky.) The frog is - of course - Hopalong, a reference to the very popular television character of the time, Hopalong Cassidy. Later, when Little Ricky names his dog, he calls him 'Fred', explaining that "I always name my pets after people I like" (Fred Mertz).
None of Little Ricky's many pets are seen or referenced prior to this episode.
When Lucy is trying to gather fish out of the bowl using a net, if you look closely at the bowl there is no fish in the bowl.