Your question "Is the predictability of the future to whatever extent is possible (based on the present and the past) equivalent to the principle of causality?" has the trivial answer ''no'' as the qualification ''to whatever extent is possible'' turns your assumption into a tautology. The tautology makes your statement false, as your question asks whether the universally true statement is equivalent to causality. An answer "true" would make any theory causal, thus making the concept meaningless.
Why is your assumption a tautology? No matter which theory one considers, the future is always predictable to precisely the extent this is possible (based on whatever knowledge one has). In particular, this is the case even in a classical relativistic theory with tachyons or in theories where antimatter moves from the future to the past.
However, in orthodox quantum mechanics and quantum field theory, causality is related to prepareability, not to predictability.
On the quantum field theory level (from which all higher levels derive), causality means that arbitrary observable operators $A$ and $B$ constructed from the fields of the QFT at points in supports $X_A$ and $X_B$ in space-time commute whenever $X_A$ and $X_B$ are causally independent, i.e., if (x_A-x_B is spacelike for arbitrary $x_A\in X_A$ and . $x_B\in X_B$.
Loosely speaking, this is equivalent to the requirement that that, at least in principle, arbitrary observables can be independently prepared in causally independent regions.
Arguments from representation theory (almost completely presented in Volume 1 of the QFT books by Weinberg) then imply that all observable fields must realize causal unitary representations of the Poincare group, i.e., representations in which the spectrum of the momentum 4-vector is timelike or lightlike.
This excludes tachyon states. While the latter may occur as unobservable unrenormalized fields in QFTs with broken symmetry, the observable fields are causal even in this case.
Informally, one major conclusion of special relativity is relativity of simultaneity. Two events that are seen (or can be deduced as) simultaneous by one person, will not necessarily be simultaneous to a moving observer. But let me try to illustrate what that really means. What happens is that their space and time coordinate axes reorient in a particular way.
For example, if you had a string of LED Christmas lights that can change color, and you set them to all simultaneously show the same color, and to cycle through colors over time, an observer moving in the direction of the stretched out string of lights will find the LEDs changing in a rainbow wave pattern. This is because the events (points in spacetime) that are simultaneous for the moving observer are "tilted" compared to your frame; the two ends of the string that they observe as contemporaneous exist at different times to you.
For clarification, take look at this image - this was made for a different purpose, but the principle is the same. In these diagrams, a physical object is always drifting somewhere "upwards" due to the passage of time. Suppose the string of Christmas lights is laid out along the horizontal axis and is simply flashing on and off (yellow bands and dark bands, respectively) as both you, the astronaut, and it go through time. The coordinate transformation for the rocket is depicted by the red dashed line. That's their coordinate grid. In it's own frame, the rocket itself is stationary (from it's own perspective), and just moves up the time axis. But look how the other axis is tilted as well; that's what's simultaneous for the rocket. See how it intersects the band? For the rocket, one end is lit (and in the past of the astronaut), and the other end is dark.
Another way to think about it is like this; suppose you print out frames of a movie and stack them in order, like you'd stack plates, to form a block (or a flip-book) that contains the entire movie. If you slice it somewhere in the middle, you'll get a particular frame. But if you slice it at an angle, you'll get a new image composed of strips of different frames. The Lorentz transformation is not exactly like that, there are some details that differ, but the simultaneity aspect is similar.
First of all I don't understand what it means that "the moment in the middle of my quarter of an hour can be considered simultaneus to your answer".
It just means that if the assumption is that it took 7.5 min to send the message one way, and then another 7.5 min to receive the response (15 min roundtrip), one could deduce that the event (point in 4D spacetime) simultaneous to the event of recipient receiving the message is when (and where) the sender was at 7.5 min mark as measured by their clock. Don't forget, we're talking about spacetime now, so when the author says "point" and depicts it in a graph, they mean a location in both space and time (it has spatial coordinates as well as a time coordinate).
But that's just relative to them. For an observer moving in a different way, the midpoint might be at a place in 4D spacetime that's not agreed by all observers as simultaneous to the reception event.
For example, for the observer in the chair, the midpoint in the second picture is in the future compared to the moment when the message arrives.
Talking about the diagram, why can we say that the observer perceives the two dots as simultaneous? [...] In these diagrams souldn't we find simultaneous events on the same horizontal line?
I think that the author was a bit imprecise with words. In "The two dots are both simultaneous to the reader, but relatively to two different motions", the author isn't saying that the reader observes the two dots as simultaneous, but that two different observers (represented by the two tilted arrows - moving through spacetime) can each consider their respective dot simultaneous to some specific event at the reader's location (so, to a third dot - "the reader is here and flips a page right at this moment"). You're right, for the reader, simultaneous points are on horizontal lines; for the two observers in motion, the simultaneity lines are tilted depending on how they move.
Best Answer
If you look at the Stanford Encyclopedia of Philosophy entry that is cited by Wikipedia, http://plato.stanford.edu/entries/spacetime-bebecome/ (search down to "Andromeda"), I think you'll see there that the argument is not especially well-accepted (that's how I interpret what's there, anyway; the SEP is usually a pretty good source, they ask good Philosophers for contributions and it's refereed by the Editorial Board). You can be sure that Philosophers argue most sides when questions of what "now" might be or of what its relevance might be are raised. I note that the original papers are in Philosophy journals.
For Physicists, the question is relevant insofar as one takes a Hamiltonian/phase space perspective or a Minkowski space-time (or other 4-Dimensional geometry) perspective, however the choice is often made lightly, depending largely on the relative convenience of the different mathematics for a particular problem.