Components of angular velocity? Let $\vec \omega = (\omega_1, \omega_2, \omega_3)$ be the angular velocity of a rigid body with respect to the body frame, where the body frame is right-handed orthonormal.
I have gathered 2 definitions of $\vec \omega$ from different sources and I am confused at how they connect to one another. One is that the rigid body rotates with $\vec \omega$ through its Center of Mass at rate $abs(\vec \omega)$. The other is that each component of $\vec \omega$ represents the rate at which the rigid body rotates about that particular basis axis of the body frame.
Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis?  
 A: 
Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis?

Yes. It's one of many consequences of Euler's rotation theorem. This neat little trick of finding the Euler rotation axis works in three dimensional space, and in three dimensional space only. The Lie group SO(3) is a very special space.
A more generic way of looking at rotation is that rotation in any Euclidean space comprises a combinations of rotations parallel to a two dimensional plane rather than about an axis. There's only one plane in two dimensional space, so rotation in 2D space requires but one parameter. There are three pairs of planes of rotation in three dimensional space (the YZ, ZX, and XY planes), so rotation in 3D space requires three parameters. You can think of those planar rotations as being about an axis (with an arbitrary decision as to what qualifies as positive or negative rotation). There are six planes of rotation in four dimensional space, so rotations in 4D space requires six parameters. This can result in something very weird, something you don't see in 2D or 3D space, a Clifford rotation.

Source: http://eusebeia.dyndns.org/4d/vis/10-rot-1
