Newton was the first to look for a source of gravity and explain equivalence of force in order to explain gravity.

No, Newton set out to explain the reason that all of the known planets were obeying Kepler's laws. For example: why do the planets sweep out an area under the arc of orbit which is proportional to time of travel?

Relativity is just the next step in first principals realized by newton to be applied to more advanced technology.

Newton's search for the explanation for Kepler's laws has nothing to do with Einstein's research, which began with explaining the invariance of the speed of light to what was at first believed to be a luminiferous aether. That research began before Einstein was born, if we mark the born-on date as the related work of Fitzeau. Einstein was not the first person to speak of reference frames and time and length contraction and dilation, but in may respects he was the final word. Special and general relativity explained this highly counterintuitive behavior of nature.

To say it carries more accuracy is no stretch for relativity. To say it carries more significance to a paradigm that it isn't related to would be another.

You would get a zero for this answer on a quiz. Kepler's laws had nothing to do with relativity, therefore, Newton's explanation, Universal Gravitation, does not account for any relativistic observations. In many cases Universal Gravitation, when used to calculate the effects of gravity, will be accurate to within more decimal places than the observer's instruments, so usually no one cares which is the better way to formulate a physical scenario. The reasons for working from Einstein's field equations usually has nothing to do with making measurements, but rather, to research other questions in particle physics and cosmology.

Relativity was engineered by talking about advanced levels of Newtonian mechanics and brownian/Gaussian motion (eg. Probability).

Zero for this too. Relativity began as research into the nature of light propagation, with no reference to either Newton or probability theory. See Einstein's 1905 paper. Also see the precedent works by Lorentz, Poincare, Michelson & Morley and Fitzeau, and any of the experiments they refer to. Also see Maxwell, Ampere, Gauss, Faraday and Coulomb for their research in electromagnetics that fomented the questions Einstein answered.

They had no time to write down every equation for every interaction until Einstein solved the basics within a field and a couple main equations.

Zero. They had no model that explained the invariance of lightspeed in all frames until Lorentz, Poincare and Einstein began to search for the answers, and began to offer explanations.

You can't derive the initial force from the acceleration without knowing the velocity.

Zero. You can't derive the initial force from the acceleration without knowing the mass.

You can't derive newtonian mechanics from relativity without probability.

Zero. You can't explain Kepler's laws of planetary motion without first deriving Newton's Universal Law of Gravitation.

You can't derive newtonian mechanics from relativity without knowing error.

Zero. It makes no sense to derive the Universal law of Gravitation from Einstein's theory of General Relativity. "Error" has nothing to do with this.

Why in this day are we trying to drive them apart?

As far as I know, all we are trying to do is to reduce the urban myths about science that are being used to leverage public opinion for the sake of deregulation and Victorian fundamentalism.

To find something of more probable than relativity and less error than newton laws?

No, to replace ignorance and fear with knowledge and opportunity.

Finding something with less error than relativity and more probability than newtons laws would be an equally difficult challenge.

Finding a good text on elementary physics and opening it is the only challenge that has not been met in this post. That, and solving the end of chapter problems.

The first principals laid down by Newton re surge in the theory of relativity.

Zero. Newton's work solved a different problem: how to explain Kepler's laws.

They are not necessarily spoken nor always mathematical, but both follow the same line in logic.

The first principles of physics are clearly explained in great detail, with copious examples and practice problems, in any typical freshman college text in physics.

All that remains to be seen is why you insist on declaring things to be so without any personal knowledge of the subject matter. Why aren't you simply asking people who took the courses to give you some advice or explanations? Why not just test yourself by enrolling in a remedial physics class?