How does Newtonian gravitation conflict with special relativity?
Newtonian gravitation is just the statement that the gravitational force between two objects obeys an inverse-square distance law, is proportional to the masses and is directed along the line that joins them. As such, it implies that the interaction between the objects is transmitted instantaneously and it must be inconsistent with special relativity (SR).
If say the Sun suddenly started moving away from the Earth at a speed very close to the speed of light, SR tells you that the Earth must still move as if the Sun were in its old position until about 8 minutes after it started moving. In contrast, Newtonian gravitation would predict an instantaneous deviation of Earth from its old orbit.
What you have discovered in your reasoning is that indeed, Coulomb's Law is NOT relativistically invariant either. But Maxwell electromagnetism is not Coulomb's Law.
As a matter of fact, Coulomb's Law is deduced from Maxwell equations as a particular case. The assumptions are those of electrostatics, namely that the magnetic field is zero and that the electric field is constant in time. These assumptions lead to the Coulomb field but they are NOT consistent with SR in the sense that they can not be valid in every reference frame since if the electric field is constant in a reference frame, then there exists another frame in which it will be varying and the magnetic field will be differnent from zero. For more you can start reading this. Maxwell's electromagnetism IS consistent with SR since the full Maxwell's equations apply in all reference frames, no matter whether the particle is moving or not.
General Relativity is the analogous for gravity of Maxwell's electromagnetism and, as it has already been said, it leads to equations for the gravitational field (the metric) analogous to those of Maxwell. Thus, it is not strange that something that resembles gravitational magnetism should appear.
It is a very interesting question that you pose and indeed that is the spirit of being a physicist. As a matter of fact there are many things wrong with that new theory you wrote and they can all be summarized by saying, 'your theory is in flat contradiction with experiment' which, of course, is what is wrong with every wrong theory.
For instance, without leaving electrostatics, your theory predicts that a static point mass gives rise to a $1/r^2$ gravitational field, that is to say, a $1/r$ gravitational potential. Therefore, your theory predicts Keplerian orbits and this we know to be not true. In the correct theory of gravitation, General Relativity, the gravitational potential of a point mass turns out to be (loosely speaking) $1/r$ plus some correction terms that go like $1/r^2$ and $1/r^3$. These terms are admittedly proportional to the probe particle's angular momentum but a particle not moving radially won't describe a conic.
The important point is that this has been beautifully confirmed by measuring to a high precission the orbits of Mercury! So, your theory must be wrong.
More importantly, your theory couples the "gravitomagnetic field" with the mass current. Therefore, classically, your theory has no effect on anything massless so your theory can't affect photons! This is again in flat contradiction with photon deflection by large masses. You might try to remedy this by coupling the theory to an energy current, for instance, instead of $\rho$ being the mass density you could take the energy density. I would need to check it but I think you would still get the wrong deflection.
Up to what extent your theory resembles gravity can be answered accurately but it would take some time. A bit more technical, your theory is actually a $U(1)$ gauge theory of gravity. Probably someone has thought of this before. One should start from the Lagrangian for both your theory and GR and find what relations may exist between your four-potential and the metric.
There is gravitational magnetic field, if you move through a static field very fast. Google for gravitomagnetism.
The main reason that general relativity is not the same is because the equivalent Gauss' law does not hold. General relativity is non-linear -- gravitational fields have energy and so act as sources for more field.