Do any planents or moons have rotation in one direction and others in another (opposite) direction?
Yes. Jupiter has dozens of moons, and at least four of them orbit in the opposite direction from most of the others. This is called retrograde motion, by the way. http://en.wikipedia.org/wiki/Retrograde_and_direct_motion
The moons of Jupiter that you would be most interested in are called Ananke, Carme, Pasiphae, and Sinope.
Neptune has a certain moon, called Triton, which fits the desciption of what you're looking for.
Saturn's moon Phoebe has a retrograde orbit.
I hope this is helpful. Keep Looking Up! --Ed