The answer to your question is the JPL horizons system.
They have a lot of documentation, but last time I checked it was pretty old school.
All the parameters you listed are given there and it is the official and most accurate data in existence AFAIK.
Kepler's laws of planetary motion make a couple of simplifications, mainly that (an artificial) satellite mass is insignificant compared to the body it is orbiting, and that there are no other gravitational bodies affecting the orbit. Other than those two major exceptions, the results would be exact.
Of course the solar system is littered with bodies that affect orbits elsewhere. These effects are called "perturbations".
In a nutshell, what the Horizons system gives you is the parameters of the orbital path and position of all the major and minor bodies at any given time, based on the effects of all the other bodies. The variation you see in the parameters such as semi-major axis and eccentricity is the result of the sum of the effects of these perturbation.
hth, I enjoy this subject, feel free to ask more questions.