When I learned about electromagnetism in my first year of undergraduate school, Maxwell's equations were derived roughly in the following way (see also here or in ): Gauss's law for a static electric field was assumed to hold in all inertial reference frames, and by carefully Lorentz-transforming to other frames you get the existence of the magnetic field, Lorentz force law, and Maxwell's equations (I hope I'm remembering this correctly. I know for sure that the former two are derived).
One consequence of this is that you get that electromagnetic waves move at the speed of light (in vacuum), and that there is no action at a distance. This is in contrast to what one might have expected from Coulomb's law, which apparently seems to work at a distance. The explanation of this is that Coulomb's law should be interpreted as the force of between charges at a static state, after there was enough time for the force to reach from each particle to the other.
1) Was an analogous treatment ever done for Newtonian gravity? If the answer is yes, than:
2) Does one also get that gravitation moves at the speed of light?
3) Is there an analog of magnetic field in Newtonian gravity? Lorentz force law?
4) Why is the resulting theory less good than Einstein gravitation?
 Purcell, Edward Mills. "Electricity and magnetism." (1965).