To what extent do the spin of smaller celestial bodies reflect the spin of the larger system’s of which they are a part? One can ask whether two objects are spinning in the same direction or not. For example, the Sun, Earth, Luna, most of our solar planets (except Venus and Uranus), and the planets’ satellites all spin in approximately the same direction. Likewise, the orbits of these bodies are in the same direction. 

I understand that the rotational plane of our solar system is tilted about 60 degrees relative to our galaxy’s. If you project one rotation onto the plane of the other, I have the impression from maps that they may have opposite orientations — i.e. if we face north to south, our solar system’s rotation is counterclockwise, and our galaxy’s rotation is clockwise. Correct? What about the rotations at other scales of the larger systems we are in, such as our local star cluster, our local galactic cluster, and so forth? Thank you.
To what extent do the spin of smaller celestial bodies reflect the spin of the larger system’s of which they are a part?
What about the rotations at other scales of the larger systems we are in, such as our local star cluster, our local galactic cluster, and so forth? Thank you.

There's an amusing explanation here: Changing the rotational rate of a natural body, it's about this much (as far as most people are concerned):

Simple and incorrect answer: In Newtonian mechanics the gravitational field of a body depends only on its mass, not on it's rotation. If you go a bit further and assume that the spinning object has perfectly uniform density then it matters not whether it spins nor the direction.
More precise answer: The Lense–Thirring effect is very small—about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive. The non-static stationary distributions of mass–energy causes frame dragging resulting in mass–energy currents and what is known as gravitomagnetism.

&quot;This approximate reformulation of gravitation as described by general relativity in the weak field limit makes an apparent field appear in a frame of reference from that of a freely moving inertial body. This apparent field may be described by two components that act respectively like the electric and magnetic fields of electromagnetism, and by analogy these are called the gravitoelectric and gravitomagnetic fields, since these arise in the same way around a mass that a moving electric charge is the source of electric and magnetic fields. The main consequence of the gravitomagnetic field, or velocity-dependent acceleration, is that a moving object near a massive rotating object will experience acceleration not predicted by a purely Newtonian (gravitoelectric) gravity field. More subtle predictions, such as induced rotation of a falling object and precession of a spinning object are among the last basic predictions of general relativity to be directly tested.&quot;.

The equations for calculating the effect are straightforward but perhaps more than you wanted to know. Modelling this complex behaviour as a curved spacetime problem has yet to be done and is believed to be very difficult.
If you're trying to point a camera at a distant object it's a big deal: