Is a falling cat's angular momentum conserved?

To change angular momentum, a torque must be applied. Since gravity pulls every part of the cat with a force proportional to its mass (that is, with the same acceleration), there is no net torque on the free falling cat, and thus no change in angular momentum.

This is true for any free falling object, but not necessarily if it is supported at any point. The support together with the gravitational force can apply a torque and therefore change angular momentum.

As to how the cat manages to turn around even with no net torque, this is known as the Falling cat problem, and is visualized nicely in this very disturbing animation from Wikipedia

rotating cat

The rotation is based on the fact that the cat is not a rigid body, and can thus bend in a way that results in its reorientation.