They are all hares, not rabbits.
Bugs Bunny and Brer Rabbit are both modelled on North American jack rabbits, which are long-eared and long-legged hares.
For years I shot at hares out in the oilfield, but rarely hit one on the run. They can run 48 mph and can leap 8 feet into the air. Chubby, my coyote, could run one down in his youth and eat it on the lawn.
Bugs Bunny, who won the oscar in 1958 for Knightly Knight, made his screen debut in 1938 in Porky's Hare Hunt. Mel Blanc, the voice of Bugs Bunny, loathed carrots: nevertheless he still had to chew them during recordings as no other vegetable produced the desired crunch.
The origins of Brer Rabbit are in the story-telling traditions of African American slaves, who told tales about the hare being more wily than the fox. Robert Roosevelt, uncle of President Theodore and friend of Oscar Wilde, was the first person to write down the stories but it wasn't until 1879 that The Uncle Remus Stories, transcribed by Joel Chandler Harris, became national classics.
The insufferably cute Easter Bunny, is also an American invention. It is a commercial sanitization of the hare as a fertility-moon symbol. In Saxon culture, the hare was sacred to Eostre, the goddess of spring, which is where we get the word Easter.
Few animals have such mythological association. From ancient Egypt and ancient Mesopotamia through to India, Africa, China, and Western Europe, hares have been portrayed as sacred, evil, wise, destructive, clever, and almost always sexy.
It's their astonishing fertility: a female hare (doe) can produce forty-two leverets in a single year. Pliny the Elder believed eating hare would make you sexually attractive for up to nine days.
The Book of General Ignorance (Everything You Think You Know Is Wrong)
If you want a wonderful tale involving a rabbit and not a hare, check out Rene Zellweger in Miss Potter.