Ant Hill Harry's tattoo reads "Maisie Singapore 1932". If we assume the cartoon's release date of 1954 to also be the date in which the story occurs, and since Ant Hill Harry's age in the TV alert is given as 35, this means he would have been 13 years old when the tattoo was inked. On the same assumption, if not paroled before then, he will be released in 2053, at the age of 134.
The music heard over the title cards is an original composition by Milt Franklyn. It would later be re-used as the title cards for Rocket-bye Baby (1956) and Greedy for Tweety (1957).
A detailed view of Bugs' home is shown here. Bugs has a TV, a washing machine, a modern bathroom...and an electric razor.
Finster's cradle reads "Baby Bugsy."
The title might be a pun on the popular children's repetition saying, Rubber Baby Buggy Bumpers.