When one sits in the sitting position, the rotational inertia of the body decreases. Since no external torque acts on the swing during this, the change of angular momentum must be conserved. To ensure this, the decrease in rotational inertia of the body must be compensated by increase in angular velocity, and the increased angular velocity eventually pumps up the swing.

The initial angular momentum due to the spin in the football ensures that any angular momentum due to the action of external forces does not change the direction of the initial angular momentum significantly. This allows the ball to reach the catcher with the same orientation as it was when thrown by the passer.
In case of the arrow, the lack of initial spin allows external forces such as gravity to change the direction of the angular momentum vector. This is accomplished by moving the arrow...

The angular speed of the student will increase. When the dumbbells are dropped on the floor, no external torque acts on the student and the turntable, and therefore the angular momentum must be conserved.
When the dumbbells are dropped, the rotational inertia of the student and turntable decreased. To compensate for the decrease in rotational inertia the angular speed of the student and the turntable must increase.

Using the parallel axis theorem

We note the angular speed and also the system energy. Using these information we calculate the moment of inertia of irregular shaped body.

Io=2E/w²

When we have moment of inertia of irregular shaped body, we can then use parallel axis theorem to solve the moment of inertia if the body about the given axis.

I=Io+md²