The trickster is a widespread (if not universal) motif in international folklore. We won't say "universal" because as soon as we use that word, someone will find an exception! But there is no question that the role of the trickster can be found in many cultures - many of which seem not to be historically related. From this, we can conclude that the trickster is either ancient or is striking a chord that is practically universal in humanity (both of these are possible if not likely).

There is no question that Bugs Bunny plays the role of a classic trickster. How much of that role is borrowed and how much of it is intuitive is difficult to say. Joe Adamson in his Bugs Bunny: Fifty Years and Only One Grey Hare (1990) sees inspiration coming from the trickster rabbit of central and eastern Africa. He also suggests that the creators of Bugs borrowed from Charlie Chaplin and Groucho Marx, both of whom fulfill their own roles as tricksters. Again, there is always the question of how much of this is intuitive and how much of it is borrowed. That said, the influence is apparent as far as Adamson is concerned, and I see no reason to question his conclusions.

It is also difficult to avoid thinking of Brer Rabbit, the Southern trickster who borrowed heavily from African culture (with a bit, perhaps, borrowed from North American Indian coyote tradition). Bugs Bunny is a media adaptation of a folk motif. He figured in well over one hundred cartoons during his first two decades after premiering in 1940 (although he has roots that date back to at least 1938). Over the expanse of that many stories, inspiration no doubt borrowed from everything the creators could grab. Was Bugs Bunny influenced by many entities as well as shared modern icons of media? Of course. Bugs is extremely clever. One would not expect him to limit the scope of his borrowings!

edit: thanks for the gold! Much appreciated. Reddit has failed to notify me through the normal means, so I can't thank my benefactor properly. Sorry!