Given an object's rotation over a given time period I would like to calculate its angular velocity. Here is what I think I need to do:

1) Scale the quaternion that represents the rotation of the object by the period of time over which the rotation occurred.

2) Convert the quaternion to a Vector3 angular velocity.

The documentation doesn't specify how angular velocity is represented other than to show that it is a Vector3. I suspect that the vector's direction represents the axis of rotation of the object and its magnitude represents the rate of rotation.

On the other hand the Vector3 returned by quaternion.eulerAngles contain rotations in degrees not a vector representing an axis of rotation. I believe that there is a way to get from the quaternion to the angular velocity vector but I don't know the right approach.

The problem is that scaling the quaternion won't work because it will no longer be a rotation (it has to be unit length to represent a rotation). The Slerp function does something similar to what you want and is a clue. Here is Slerp:

Slerp(q0, q1, t) = q(t) = (q1 q0*)^t q0
(q0* is Conj[q0], which is the inverse for unit length quaternions)

This interpolates between two rotations q0 and q1 with t varying from [0, 1].
So if you think of t as being time, then the time derivative dq/dt is:

Log[q1 q0*] (q1 q0*)^t q0

also,

dq/dt = 1/2 w(t) q(t)

Note the equivalent form, the Log[q1 q0*] part of dq/dt matches with 1/2 w(t), so the angular velocity is:

2 Log[q1 q0*]

For unit length quaternions this evaluates to

2 v / ||v|| ArcCos(s)

where quaterion q is (s, vx, vy, vz) (i.e. s is the scalar part and v the vector part)