If the solar system is a non-inertial frame, why can Newton's Laws predict motion? Since there is no object in the universe that doesn't move, and the solar system likely accelerates through space, how did Newton's Laws work so well? Didn't he assume that the sun is the acceleration-less center of the universe? Shouldn't there be many psuedo-forces to account for planetary motion?
 A: 
Shouldn't there be many psuedo-forces to account for planetary motion?

In theory, yes. In practice, no.
Consider the third body perturbations induced by Alpha Centauri (a two solar mass star system at a distance of 4.37 light years) on Voyager 1, which is currently about 130 astronomical units from the solar system barycenter. This is on the order of 10-16 m/s2. The result of this tiny of an acceleration is completely unobservable, even over a long span of time. The third body perturbations by the galaxy as a whole is almost an order of magnitude smaller that the perturbations induced by Alpha Centauri.
By way of comparison, the tiny Pioneer anomaly, now attributable to asymmetric thermal radiation, is roughly seven orders of magnitude larger than the perturbations induced by Alpha Centauri. Those extra-solar third body perturbations are so very, very small that they are essentially unobservable.
One possible exception would be a star that comes closer to the Sun than 4.37 light years and perturbs the orbit of an object that orbits the Sun well beyond 130 astronomical units. Another name for such a perturbation is "long period Oort cloud comet." Even then, it will take millions of years for those perturbations to take hold.
A: 
Since there is no object in the universe that doesn't move, and the solar system likely accelerates through space, how did Newton's Laws work so well? Didn't he assume that the sun is the acceleration-less center of the universe? Shouldn't there be many psuedo-forces to account for planetary motion?

Newton assumed his laws were valid with respect to absolute space (special reference frame, something like all-pervading solid body that does not hamper rectilinear motion of other bodies). He did assume that Sun moves with negligible acceleration with respect to this absolute space. And it worked well.
Today we do not think the idea of absolute space is that necessary, but simply apply Newton's laws with respect to some reference frame (Earth, solar frame, galactic frame...) and see if the description corresponds well to actual motions. If it does, we say the reference frame is inertial enough. If it does not, we say the frame is not inertial enough and either seek another frame or introduce pseudo-forces.
Often the frame of the most massive body in the system of interest with fixed orientation with respect to distant stars is inertial enough. If the bodies have not that different masses or the frame is not inertial enough for other reasons, we may try the frame of the center of mass of the system or seek another frame until the laws apply well.
In the case of solar frame S, it works well and no pseudo-forces are usually needed. This does not mean the solar frame does not accelerate with respect to some other frame G, only that acceleration of the bodies of the solar system with respect to G is so uniform across the solar system that it can be neglected in the frame S.
A: There's no doubt the solar system is accelerating. The milky way galaxy rotates, and we're quite on the outside. Hence, there's a permanent acceleration vector pointing to the center.
However, this is a phenomenally small acceleration. If you'd try to measure it here on earth, you run into all kind of practical problems when you try to isolate it. For instance, the earth's gravity isn't really that constant, at this scale. The tides move ocean water around, in reaction to the moon's gravity. 
So, in practice, when Newton's law is a sufficiently good approximation (relativistic effects small enough), the Sun can be considered to stand still.
A: There are two main reasons it is practical to ignore the pseudo forces due to the rotation of the earth/sun about the galaxy. First, the accelerations are pretty small, and second, they are pretty uniform.
The sun moves around galactic center at about 800,000 kilometers per hour, but it takes around 250 million years to complete a single orbit of galactic center.
Using $v=2\pi r/T$ we get $r=vT/2\pi$.
So for a circle $a=v^2/r=v2\pi/T\approx 2\times 10^{-10} m/s^2$ which is pretty small.
The other factor is that the acceleration is pretty uniform.  Tidal forces fall off like $1/r^3$ instead of $1/r^2$ so they are even smaller for large distances.
A: Newton's laws work well but if one considers the relativity theory, one finds things not explained by Newton's laws. A well-known example is the "anomalous" precession of the perihelion of Mercury, explained by the general relativity. 
A: When 1686 Newton writes "Principia...", the inertial frame concept does not exist yet. However, we can find in it Corollary IV (introducing the center of mass CM concept for any interacting body set), Corollary V (Galileo's Principle of Relativity, applied to any limited body set with CM at any uniform velocity), and the today almost forgot Corollary VI (a generalization of the V from CM acceleration zero to any variable one). The application of Corollary VI to the Solar System determines that all occur inside it (as if it were a Galileo's ship) in the same way (same 1686 Newton's Laws and other natural ones), no matter at all its CM acceleration, known or not.
Rafael A. Valls Hidalgo-Gato; Institute of Cybernetics, Mathematics and Physics; Havana, Cuba.
