This question is more about physics than about math, so I hope you will bear with a physicist for posting an answer. The OP expressed the following doubt:
...Of course, at these examples, the speed of these curves increase infinitely ("goes to infinity"). ... what axioms do I insert to my models in order to get such results, from a purely mathematical/axiomatic point of view? Is the answer to this "the Einstein postulates"?
or, in other words, is there an extra requirement that space-time manifolds ought to satisfy, as compared to generic manifolds with the same signature as the Lorenzian metric, in order to prevent the existence of superluminal motions?
If this is indeed your worry, then no, there are no other requirements which distinguish a space-time manifold from another manifold (with the same signature): it is a matter of interpretation.
Since the metric is not positive-definite, there will be geodesics with $ds^2 = 0$, $ds^2 > 0$, $ds^2 < 0$. The geodesics with $ds^2 = 0$ (called null geodesics) describe the motion of photons. An interval $ds^2 < 0$ can always be rewritten, thru a change of coordinates, as $-dt^2 < 0$, which shows the separation to be time-like, and intervals with $ds^2 > 0$ can likewise always be expressed as $dx^2 > 0$, which shows the interval to be space-like. Now, the superluminal curves you refer to do exist in a space-time manifold: they are the curves on which, at least somewhere, $ds^2 > 0$, but they are not interpreted as the space-time motion of any physical observer, exactly because they are superluminal. Only time-like geodesics, i.e. those with $ds^2 < 0$, are possible paths for physical, subluminal observers.
A graphical way to represent this is to consider the bundle of null geodesics (both future-oriented and past-oriented) going thru a given space-time point $P$. This bundle is composed of two hypercones (the future-oriented null geodesics, and the past-oriented null geodesics) with vertices joining at $P$. All points within the cones may be joined to $P$ by a time-like (hence, subluminal) curve, thus providing a nice, coordinate independent characterization of the past and future of $P$.
Points lying outside both cones are not causally related to $P$,because they require superluminal motions to be connected to it. But they do play an important role in physics: a 3-dimensional hypersurface thru $P$ but lying otherwise wholly outside the null cones of all of its points provides an adequate surface for setting up boundary conditions for a time-dependent problem (like the evolution of the Universe) for obvious reasons of causality.
As for reading suggestions, there are at least two books by physicists which I consider eminently suitable to mathematicians: first, The large scale structure of spacetime by Hawking and Ellis, and then General Relativity by R. Wald. Both are classic texts; that by H&E focuses on on the study of singularities, while the one by Wald is wider in scope. But H&E also contains very interesting discussions about the properties of special solutions (like Taub-NUT space, or Godel's rotating Universe) which will at some point enjoy a comeback, IMHO. Basically the two, which have a large amount of material in common, complement each other very well in the remaining parts.