Does Earth's rotation affect the orbital velocity of its satellites? We all know about the spin of black holes. That spin and of course it's gravity determines the minimum distance a star can orbit the black hole with a min orbital velocity.
Do we see similar result? Does spin of earth also effect the orbital velocity of its satellites?
 A: The effect of black holes on prograde and retrograde orbits is due to frame dragging, which is a general relativistic effect. Hence the effect is microscopic on Earth satellites.
The LAGEOS satellites tried to detect frame dragging around Earth, but did not succeeded directly. NASA has claimed success with GRACE, though. A more recent paper used LAGEOS and other satellites to find the effect. The drift due to the effect is about 30.68 milli-second of arc per year, so it is totally minuscule.
The big source of noise in these measurements is tidal effects due to the lumpiness of Earth, which is also changing over time due to other bodies. That brings up a bigger, classical effect on satellites that does matter in practice. A rotating planet with a tidal bulge will pull and push on the orbiting satellite depending on how the orbital period fits with the planetary rotation.
There is also a difference in how easy it is to capture natural satellites in prograde and retrograde orbits (classic paper). Retrograde orbits average the bulginess more, and are somewhat more stable, while prograde orbits tend to change more.
For big enough satellites (or rather moons), the bulges they induce themselves matter. If they are above prograde geosynchronous orbit the bulge will tend to be ahead of them (Earth rotates faster than the moon orbits) and pull them ahead, moving them into a higher orbit and slowing Earth's rotation. The opposite would happen in a retrograde orbit. But this is all classical physics.
