Along the idea of frame-dragging;

Will the rotation of a black hole, which has some velocity v and angular momentum, influence its path in 3D space?

I've seen the fact that depending on the direction of the rotational velocity, the Kerr metric can give different revolution periods for orbiting masses. (e.g. as in our solar system)

So why (or why not) is this so?

How does the metric show that the direction of the velocity can influence orbital periods?