Yes. Of course, you would need a fixed pivot about which to work and that might be hard to find. But you could do work on the world with your lever. If the arm you were dealing with was long enough, you could do that work with a small force exerted over a very, very long distance. The lever would then do this work on the world with a very, very large force exerted over a small distance.
Cats manage to twist themselves around by exerting torques within their own bodies. They aren’t rigid, so that one half of the cat can exert a torque on the other half and vice versa. Even though the overall cat doesn’t change its rotation, parts of the cat change their individual rotations and the cat manages to reorient itself. It goes from not rotating but upside down to not rotating but right side up. Overall, it never had any angular velocity. As for skydiving, that is mostly a matter of torques from the air. As you fall, the air pushes on you and can exert torques on you about your center of mass. The result is rotation.
No, moment of inertia embodies both mass and its distribution about the axis of rotation. The more of the mass that is located far from the axis of rotation, the larger the moment of inertia. For example, a ball of dough is much easier to spin than a disk-shaped pizza, because the latter has its mass far from the axis of rotation.