Does Earth-spin affect the retrograde orbital motion of artificial satellites? If frame-dragging is true then how the retrograde satellites orbit in the opposite direction to the earth spin direction?
 A: Since you've tagged your question "General Relativity", I'll assume you're talking about the Lense Thirring effect aka "Frame Dragging".
If so, your question bespeaks maybe some confusion about how the phenomenon affects orbitting bodies. You seem to imply that the effect literally "drags backwards" against the retrograde orbitting object's tangential motion. Indeed, the very word "Dragging" in "Frame Dragging" betokens a kind of "viscosity". This is not how it works, however. Instead, the frame of an observer orbitting the body is entrained to spin in the opposite direction from the the spin of the central gravitating body. There is a particular angular speed, relative to distant stars and counter to the rotating gravitator, at which the there are zero "torques" on the body and its spin will stay constant. This state of spin relative to the distant stars is, for the object, its local "freefall" or "nonrotating" state. The present Frame Dragging Wikipedia Page page does a pretty good job of putting this into words:

This produces interesting locally rotating frames. For example, imagine that a north-south–oriented ice skater, in orbit over the equator of a black hole and rotationally at rest with respect to the stars, extends her arms. The arm extended toward the black hole will be "torqued" spinward due to gravitomagnetic induction ("torqued" is in quotes because gravitational effects are not considered "forces" under GR). Likewise the arm extended away from the black hole will be torqued anti-spinward. She will therefore be rotationally sped up, in a counter-rotating sense to the black hole. This is the opposite of what happens in everyday experience. There exists a particular rotation rate that, should she be initially rotating at that rate when she extends her arms, inertial effects and frame-dragging effects will balance and her rate of rotation will not change. 

The particular rotation rate in question is, with reference to the "Mathematical Derivation of Frame Dragging" section further down the same Wiki page:
$$\Omega = \frac{4\,G\,M\,\omega }{5\, c^2\, r}\tag{1}$$
where $\omega$ is the gravitating body's spin rate, $M$ its total mass and $r$ the orbital radius. I get this expression in the large $r$ limit for the ratio $g_{t\,\phi}/g_{t\,t}$ of the two metric co-efficients $g_{t\,\phi}$ and $g_{t\,t}$ when $r\gg r_s,\, \alpha$. For the Earth, the Schwarzschild radius $r_s$ is about a centimeter, the $\alpha = J/(M\,c)$ paramter ($J$ is the Earth's angular momentum) is about $3.7{\rm m}$ compared with $r=6500{\rm km}$ for LEO - hence my approximation.
The precession rate in (1) is TINY, working out to about $7.2\times10^{-4}$ arc minutes a day (or about a degree every two thousand years!). It's not the only relativistic precession effect, but its one of the largest. De Sitter precession is comparable, and special relativistic Thomas Precession, at the LEO speed of $7{\rm km s^{-1}}$, is two orders of magnitude smaller. There are two simulators that you can explore these other effects with on the Wolfram Demonstrations Project; see
Thomas Müller, "Geodesic Precession on a Timelike Circular Orbit around a Schwarzschild Black Hole", Wolfram Demonstrations Project
for de Sitter precession and my own
Rod Vance, "Thomas Precession in Accelerated Planar Motion", Wolfram Demonstrations Project Published: September 29, 2015
for Thomas Precession.
The same precession rate as in (1) applies to other, related, phenomena. For example, (1) defines the precession rate (relative to the distant stars) of the plane of a Foucault Pendulum swinging over the North or South pole. One adds a colattitude $\cos\theta$ term for non polar positions. It is also the rate of precession of the Nodal Line of a star in an orbit whose plane is tilted relative  to the gravitator's rotation plane. Alternatively, if the orbitter's plane and gravitator's spin plane are the same, and the orbit is elliptical (1) becomes  
$$\Omega = \frac{4\,G\,M\,\omega }{5\, c^2\, r\,(1-e^2)}\tag{2}$$
where $e$ is the orbit's eccentricity, and the frame dragging effect on the orbit is qualitatively similar to, and further to, the apsidal precession that a nonrotating gravitator begets (as first observed on Mercury's orbit). It is hoped that astronomers will confirm the Lense Thirring effect on the orbits of the stars that orbit our galaxy's central black hole in the coming years: you probably know that these are all highly eccentric orbits lasting about a decade and thus, from (2), that they are highly susceptible to the effect.
To pull back a little and become more practical: as you can see, these relativistic effects are utterly tiny, and other orbital destabilizations are likely to be much more important and noticeable. For instance, density inhomogeneities (mass concentrations) in the Earth destabilize orbits so that satellites need to use fuel constantly to hold a stable orbit - this is the main reason for their limited lifetime. I do not know the quantitative details of this assertion; to seek them would be a good question to put to Space Exploration Stack Exchange.
