Because it rotates in situ, its center of mass does not move, so it is static, but it is rotating, so it is not static, then is it static or moving?
It depends on your definition of moving. It seems like you aren't picking one, so you are confused.
If you define moving to be any part of the object has motion relative to you, and so then stationary means no part of the object is in motion relative to you, then you would say a rotating object is moving and is not stationary.
If you define moving to be the center of mass has motion relative to you, and so stationary means the center of mass has no motion relative to you, then an object that is purely rotating about it's center of mass would be considered stationary.
I can think of instances where either definition could be useful, so pick your favorite.
As you pointed out, if an object is rotating around an axis through its centre of mass, then the centre of mass is stationary. Nonetheless, you will not find an inertial frame of reference where all particles of the extended object are at rest to each other.
All the particles of the rigid body that are not located on the axis of rotation would move even if the centre of mass remained stationary. This is also the reason why one often divides the motion of a body into its translational and rotational modes. For the translation of a rigid body it is sufficient to look at the centre of mass only, this is not the case for rotations.
It is moving, as it has kinetic energy. The center of mass is a point, not an actual measurable piece of mass. Think of a donut spinning like a wheel, you would see the entire donut rotating, and there would be nothing of the donut in it's center of mass, the middle of the donut hole.