The reason why there are so many "adult" jokes in Bug Bunny is simple. When they were originally made 1930s/1940s (at least the classic ones) they were not made for kids. They were shown at the begining of movies with the newsreels (which many times were about WW2, so also not for kids). The same is true for the old Tom & Jerry cartoons. They had adult jokes because that was the target audience. Keep in mind that, at the time, animation was a relatively new technology (for mass audiences that is), and hadn't yet gotten the stigma of being "just for kids".

How and why these cartoons, originally intended for adult audiences, made their way to stock "kids" programming is a whole other issue. If you watch, for instance Tom & Jerry, in chronological order (yes I have done that, my friend got 15 VCD of them from Taiwan and we watched just about all of them). You will notice the decline in "adult" humor as the years go by, you will also see the decline in racist "humor" (Tom's original owner was very much a "Mammy" stereotype) and WW2 references (there were many in early Bugs Bunny too, and even some WW2 propoganda films featuring Bugs).

As for the Flinstones, it was a blatent rip-off of the Honeymooners, and originally aired in prime-time back in the 1960s. Now we may not think of prime-time as an adult time-slot nowadays, but back then, it was. Sure some kids saw it, but they (again) were not the target audience. Its humor, and storylines were very much oriented towards adults, this can be seen best in the Winston cigarette ad featuring Fred and Barney. You can find a Real Video stream linked of this page, or the direct Real Video stream here. Be sure to listen to the whole thing, at the end the announcer says "The Flinstones were brought to you by Winston".

Just because its a cartoon, doens't mean its only for kids. Any Manga/Anime fan will tell you that.