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.
I think I understand what you're asking so I'll answer accordingly. Ignore this answer if I've got the wrong end of the stick.
General relativity tells us that the four acceleration is given by:
$$ A^\alpha = \frac{\mathrm d^2x^\alpha}{\mathrm d\tau^2} + \Gamma^\alpha_{\,\,\mu\nu}U^\mu U^\nu \tag{1} $$
So there are two contributions, the time dependence of the coordinates and the term in the Christoffel symbols. Since the four-acceleration is a four-vector the norm of the four-acceleration, the proper acceleration, is an invariant so it will be the same in all coordinate systems.
If we consider a freely falling observer in Minkowski spacetime (i.e. your lift) then the norm of the four-acceleration is zero. As you say, we can choose coordinates where $\mathrm d^2x^\alpha/\mathrm d\tau^2=0$ and $\Gamma^\alpha_{\,\,\mu\nu}=0$ and this is what we'd call an inertial frame. Alternatively we could choose accelerating coordinates, like the Rindler coordinates, where neither $\mathrm d^2x^\alpha/\mathrm d\tau^2=0$ nor $\Gamma^\alpha_{\,\,\mu\nu}=0$ but of course the proper acceleration of our freely falling observer would still come out as zero.
I'd guess we agree so far, but where we disagree is that I don't see that there's anything different between GR and SR or indeed classical mechanics. The invariant is the proper acceleration of the observer and that is always unambiguously measurable because the observer just has to weight themselves. The same equation (1) applies to curved spacetime, flat spacetime and indeed to non-relativistic motion where the manifold is Riemannian.
Best Answer
Yes, special relativity is a special case of general relativity. General relativity reduces to special relativity, in the special case of a flat spacetime. I.e., general relativity reduces to special relativity, in the special case of gravity being negligible, for example in space far from any objects, or when considering a small enough piece of space in freefall that gravity is unimportant to the problem.
Like special relativity, general relativity also assumes that the speed of light is universal. However, when spacetime is curved, the universality of the speed of light can only be applied locally, within regions of spacetime that are small enough that the effects of gravity aren't important within the region.