There are a few ways to get angular momentum!
For a rotating rigid body, you can get L (angular momentum) from the moment of inertia (I) and the angular velocity (omega): L = Iw
For a point mass, you can get angular momentum straight from the linear momentum §: L = p R = mvR
You can also get your change in angular momentum from the torque on the object over a period of time (the rotational version of impulse!): change in L = torque * change in time
To clarify here, L is the angular momentum. I think you’re mixing up your quantities and symbols. L is angular momentum, omega (the swirly w) is the angular velocity.
Also for your last point, angular momentum is conserved. When the ballerina pulls her hands closer to her body, she is decreasing her moment of inertia (moments of inertia are typically larger if the mass is further from the axis of rotation). From L = I w, if I goes down, w has to increase to compensate in order to keep L constant.