Okay, we're getting nowhere so lets see if we can clear a couple of points up:
1) 'Now' has nothing to do with how much time has passed. That's kind of the point. Let's use one of Einstein's examples: Take a disc shaped plane. Draw a radius and label the outer point A and the inner point B. Now, spin the disc in its own plane at a significant fraction of the speed of light. In accordance with the predictions of Lorentz, a clock at point A will always experience time more slowly than a clock at point B. Further a measuring rod tangent to A will always be shortened in the direction of the spin (as a fun consequence the ratio between the circumference of the disc and the diameter is no longer pi); however, the radius itself does not change. That radius is the 'now'. It's important not to get hung up into elapsed time or even the simultaneity of observed events. The basic model of a (Euclidean for special/Gaussian for general) space manifold comprising the present instant (our radius) compounded with a time parameter should always hold.
The purpose of that point is that there can be no special ordering between spacelike separated events according to relativity. Events A, B, and C, if spacelike, can happen in order A, B, C, or C, B, A, depending on the motion of the observer. Or some other ordering. What they consider to happen first or second is a matter of perspective and thus relative.
If a signal can move faster than c, you either get a causality violation or determine a special ordering.
2) Minkowski's work is the basis of our modern concept of spacetime. We have built on his model, but not supplanted it. That's like saying that Darwin's theory of evolution by selection isn't the basis of our modern evolutionary biology just because we have built on it. Or the ideal gas law. Or electrostatics. Einstein himself said that without the Minkowski space-time continuum the general theory of relativity would have "got no further than its long clothes". All that is required for a shift from special to general considerations is the conversion from a Euclidean to non-euclidean continuum (the parameters of the shift based on the apparent gravitation from your reference frame).
The Minkowski solution is used and taught because it is so simple - it's easy to do the transforms, and, like calling the Earth round instead of geoid, works for most purposes. This does not change the fact that it is not an accurate depiction of spacetime. It does not permit known FTL phenomenon, such as black holes or the expansion of spacetime, which general relativity does.
3) I care as much about ansibles as I do about rubber-forehead aliens. If you want to talk about sci-fi, fine, but it is not a response to anything I am saying. I'm talking about the implications of current science on current theories. Neutrino's moving faster than the speed of light means that they may be able to move in a Galelei-Newtonian fashion. Which means they would violate what we have defined causality (roughly speaking a Minkowski light-cone) at sufficiently differing reference frames (that we personally are in no way capable of achieving). That's not time travel or FTL travel in any conventional sense (either in the sense used by overenthusiastic geeks and the press, or in the literal sense of transversing the manifold of the present). It also does not violate the principle of relativity.
You can do the experiment with any superluminal communication medium - such as these neutrinos if the experiments were verified. The math is simply easier with ansibles as you don't have to work out what a given superluminal speed means. If you have a superluminal communication method, you by definition are able to either use relativity to break causality (because there is no special ordering between the spacelike events you are communicating between), or use the subsequent experiments to determine a special ordering and thus a special frame (thus violating a central tenet of relativity). A quarter of a percent over c is still enough for a type I or so civilization to work with, so it would make an experiment feasible at least in some idealized future (you don't have to go at .999 of c to actually create a violation, just fast enough so that the contraction can be measured and see how it applies to the neutrinos).
Since the possibility that these particles are going a noticeable fraction faster than c is the central point of our argument, your comment of lumping in a confirmation with 'rubber-forehead' aliens is nothing but pure trolling, and not appreciated. In at least one typical model of a tachyon, they get faster as they bleed energy, meaning that lower-energy neutrinos would move faster still (though they are harder to detect - the Earth is actually opaque to neutrinos on the PeV scale or so). It's the entire point.
Very few educated people expect these results to be verified. But if you are going to make the argument from the assumption that they are verified, then yes, an ansible is a mathematical toy concept that belongs in the discussion whether you care for it or not.