Well, when I was taught the Foo Foo song, waaaay back in the late '60s, it was little Rabbit Foo Foo, and he was running through the forest, scooping up the field mice and bashing them on the head.
That little extra degree of violence really helps drive home the point that being a bully is wrong. Also, rabbits are not as sympathetic as bunnies.
All in all, I think it was a much more effective song back then.
As for “It ain’t gonna rain no more,” my father, brother and I used to sing it while on long road trips, and we frequently added new verses. I have no idea what the original verses are, but some of the verses we sung can be found on this page (along with a bunch of other similar songs we used to sing).