How do you get curves in Box2D?
In the unreleased version of Box2D, there are two new shape types, b2LoopShape and b2EdgeShape, which let you build curves out of a sequence of line segments.
In the released version, you can use multiple small boxes set at angles to get a similar effect, but bodies have a tendency to do unstable and unexpected things near connection points. So I'd recommend using the unreleased version.
How can a character fall down the curve?
Make the character a wheel! My question about platformer physics has some discussion on this, but the basic idea is to mount your character's body on a rotating wheel, and move by applying rotational force to the wheel. Even though (actually, because) the wheel has friction, the downward force from gravity will cause it to rotate down slopes - like a real wheel. By changing the revolute joint's motor parameters, you can make it resist rotation as much as is necessary to avoid it rolling down small inclines.