Do all predictions of special relativity follow from the Lorentz transformations? Is there a proof that all observable predictions of special relativity follow from the Lorentz transformations?
I have edited my question with concrete examples, so that it can be understood more easily.
The formulation of special relativity I have in mind is that exposed by Einstein in 1905, before it was reformulated in term of the Minkowski spacetime. The original formulation rests on the following assumptions:


*

*The principle of relativity

*The constancy of the one-way speed of light

*Homogeneity of space and time


From these assumptions can be derived the Lorentz transformations. Then from the Lorentz transformations can be derived experimental consequences, i.e. predictions.
Consider the following Lorentz transformation between two relatively moving observers:
$$
x' = \gamma(x - v t) \\
t' = \gamma(t - \frac{v x}{c^2})
$$
By setting $$x = vt$$ one gets $$t' = \frac{t}{\gamma}$$
If the two observers were initially at the same location and synchronized ($x=x'=0$ when $t=t'=0$) then the Lorentz transformations predict that when the observer's clock at the origin shows the time $t$, the moving observer's clock shows the time $t' = \frac{t}{\gamma}$ . This is an example of a non-observable (or non-testable, non-verifiable, non-falsifiable, ...) prediction, for we have no instantaneous signals to test if this is indeed the case. 
However the relation $t' = \frac{t}{\gamma}$ leads to testable predictions. For instance it predicts that when the moving observer is at a distance $d$ from the origin, his clock will show the time $$t' = \frac{\frac{d}{v}}{\gamma}$$
It also predicts that if at time $t'$ the moving observer instantly turns around and move back towards the origin at velocity $-v$, then when the two observers meet again the moving observer's clock will show the time $2t'$, while the clock of the observer who stayed at the origin will show the time $2t'\gamma$. This is an observable prediction, that can be checked experimentally.
Similarly the Lorentz transformations allow to derive many other observable predictions, such as that a signal of frequency $f$ emitted by the moving observer towards the origin will be measured by the observer at the origin to have the frequency $f\sqrt{1+\beta\over 1-\beta}$. In other set-ups it allows to derive the angle of aberration of distant light sources.
All these observable predictions are a consequence of the assumptions of special relativity, since the Lorentz transformations are derived from them. But these predictions can also be arrived at without using the Lorentz transformations, by starting from the assumptions of the theory. So how do we know that the Lorentz transformations encompass all the possible observable predictions that can be made, starting from the assumptions of special relativity?
In other words, if we call A the assumptions of special relativity, B the Lorentz transformations, C the observable predictions derived from the Lorentz transformations, and D the observable predictions derived the assumptions of special relativity, how do we know that C is exactly the same as D and not just a subset of D? Is it possible to prove that C and D are the same?
As to the motivation behind this question, I am wondering whether two theories that start from different assumptions but from which Lorentz transformations can be derived are necessarily indistinguishable experimentally. My point of view is that if the Lorentz transformations do not encompass all the predictions that can be derived from the assumptions of each theory, then in principle the two theories may be distinguishable.
 A: Special relativity is the spacetime geometry described by the Minkowski metric. It is the vacuum solution to Einstein's equation with the lowest ADM energy. All the properties of the geometry, time dilation, lorentz contraction, etc, are described by this metric.
Starting with the Minkowski metric it is possible to derive the Lorentz transformations. They are a special case where all motion is unaccelerated. If you wish to describe accelerated motion you need to go back to using the metric directly.
So there is no:

proof that all experimentally verifiable predictions of special relativity follow from the Lorentz transformations

because they do not - they are a special case. As for a proof that all verifiable predictions of SR are described by the Minkowski metric, that's a tautology because the Minkowski metric is special relativity.
A: Historically, Lorentz transformations were discovered before Einstein's relativity. But only special relativity with its 2 postulates formed today's theoretical base for Lorentz transforms. So, today there is no doubt that Lorentz transforms may be derived from SR.
Now you are asking if it would be possible to derive conversely the two SR postulates from Lorentz transforms. This is not the case, because SR postulates imply certain constellations which are not considered by Lorentz transforms. In particular, certain parts of Minkowski spacetime such as the light cone, zero spacetime intervals and lightlike movements are not taken into consideration by Lorentz transforms, the transformations being limited to reference frames.
Edit/ Observable predictions: The movement at speed of light cannot be predicted because Lorentz transforms are referring only to reference frames, and massless particles do not have any reference frame.
A: The Lorentz transformations by themselves have no physical content and make no predictions. They basically take things called frames, each of which contains some ordered four tuples of numbers, and describes a particular bijection between the set of four tuple in one fixed frame and the set of four tuples in another fixed frame.
If you wanted to, you could do all of SR without ever picking or using a frame. You could have a flat homogeneous isotropic manifold with a metric of Lorentzian signature.

This is an example of a non-observable (or non-testable, non-verifiable, non-falsifiable, ...) prediction, for we have no instantaneous signals to test if this is indeed the case.

It's an oxymoron to say you have a non-falsifiable prediction. A prediction is a precise statement of something that could be observed to happen and could also be not observed to happen.
In this case you'd need to use SR to make a model. The model could be that for a each inertial frame, clocks can exist at rest in that frame and the inertially moving clocks measure time durations between two events where the clock is situated according to the time coordinate difference for the four tuples corresponding to the two events.
The model required that you correspond events in reality to events in the mathematical model. The mathematical model could be a 4d manifold. It could also be a 4d vector space. Each real event could also be identified with an equivalence class of four tuples of coordinates, one four tuples from each frame, specifically chosen so the transformation maps the corresponding four tuples to each other.
You could even start with the manifold and then select frames as different assignments of four coordinates to each 4d point on the 4d manifold. Which you start with, and even whether you choose to use Lorentz transformations is up to you.
Then you can make a synchronization convention so that a bunch of inertially moving clocks all with zero relative velocity (which can also be defined in the framework of SR) all basically record the time coordinate of that frame in which they are at rest.
So now you simply take a clock at rest in each frame that record 0 when both are the first event. And take another pair of clocks at rest in each frame that record $t$ and $t'$ when both are at the second event.
Since clocks record times and since the second pair are both at one and the same event, the comparison is directly achievable and the record can be sent to the future for analysis.
SR has a common future for any events, so its actually one of the features that any fact that can be recorded and transmitted in two regions can be analyzed jointly and together in some region. So if you think otherwise you've failed to grasp which models the theory allows.
Also keep in mind that the Lorentz Transformations are basically maps of vector spaces. General Relativity can use them for the set of tangent vectors at a point. And Special Relativity could use them on event themselves. But those are totally different uses of the same mathematics and the same symmetry.


what if I characterize Lorentz transformations by: "If by the assumptions of SR A infers an event to happen at (x,y,z,t) and B infers an event to happen at (x',y',z',t'), then the relations between (x,y,z,t) and (x',y',z',t') are called the Lorentz transformations", would you agree then that these relations allows to make predictions? 

Firstly, no those still make no predictions. All you are doing is identifying a set of labels for one person with a set of labels for another person. And secondly you should use Poincaré Transformations for SR. Lorentz transformations always identify (0,0,0,0) for one person and (0,0,0,0) for another person. But even then, still you have no predictions.
Someone could identify points from different people's vector spaces and then not use the vector space to make predictions. Even if they used them, they could have equations like $F=ma$ or equations like $E=mv$ or $G=mj$ ($j$ for jerk) there is no rule that they have mass, inertial motion, momentum, second order differential equations for dynamics. And in fact Maxwell doesn't have mass for electromagnetic fields and does have first order dynamics. And relativistic fluid mechanics and particle mechanics does have mass and has second order dynamics. So there is at least two types, and they are coupled.
Really you should think of SR as either a meta theory telling what symmetries your actual theory should respect. Or else you need way more to it than just giving a bunch of people some vector spaces and identifying points in them.
Without either, you could ignore the vector space and instead use something else to make all your predictions and your predictions could be arbitrarily different than physics as we know it. Predictions come from models. Models 

in order to synchronize the clocks one has to assume that light travels at c in all directions.

That isn't actually true. Clock synchronization is basically a convention. If light "really did" move at different speeds in two directions, we could still use the radar time convention to synchronize clocks and postulate that a clock moving inertially between two events that have the same spatial coordinates in some frame measures and records the time coordinate difference in that frame where the spatial coordinates don't change. Then you've connected clocks to you coordinates. And the synchronization allows clocks that are mutually at rest to 1) be identified as mutually at rest 2) to be identified as synchronized if synchronized according to the convention. This making of a model required way way way more than just having lots of vector spaces and identifying one vector from each with an event. We have to connect aspects of the mathematics to thibgs that can be made, done, seen, and observed in the real world.
For instance we can connect clock readings to time coordinates. We could also connect ruler readings to spatial coordinates. We can say that objects track out differential (or at least connected) parameterized curves in the vector spaces, or in the spacetime and call those parameterized curves worldlines. This tells us that particles can move in the spacetime (we didn't have that before, the LT by itself didn't tell us that). We could assert that some move so that only a time coordinate changes, and we could call those worldlines inertial motion. And we could say those particles have mass and we can assert that every frame agrees it has the same mass and that it never changes. We can assert that some particles have their x and t coordinates change an equal amount. We can assert that those particles have an energy-momentum four-vector that points in the same direction as the tangent to the worldline. We could say that the massive particles don't have to move inertially (though the inertially moving ones do have to have mass).
Obviously the LT by themselves do not tell us that because they don't mention mass or momentum.
Even in introductory physics you probably remember the section on kinematics and then dybamkcs came later. Kinematics didn't make predictions. The LT don't make any predictions either. They merely connect some vector spaces for different people. I'm flabbergasted that you can even imagine the LT making any predictions. It's like if some people named their cats and someone else named the cats their own names. The Lorentz Transformations could tell you which cat names the first person uses relate to which cat names the second person uses. But it doesn't matter. Cats don't respond to their names (and if you think they do use fish names or something else instead). Identifying labels doesn't make predictions unless those labels are used to make predictions.
You need dynamics to make predictions. Something like Maxwell. Or something like $F=ma$ and some force laws. Or something about how clock readings are related to time coordinates. Or something about how ruler readings are connected to x,y, and Z coordinates. Or lots of those.

along with the assumption of the constancy of the speed of light which is implied in order to apply the relations experimentally

If you have the LT and you relate the time coordinates to clocks and the spatial coordinates to rulers, then you really think you need to add an extra assumption of constancy of speed c? That doesn't even make sense. You can't just add an assumption to an existing mathematical framework and not change the framework and expect that you are using it. The closest you can get to that is like Newton's third law where you decide to restrict your changes in momentum so that when something loses some momentum it gives an amount equal momentum what it lost to something else. Before you had that law you were allowed to do otherwise, and you placed a restriction on your allowed models to not do that. Or with Maxwell you could asd the equation $\vec \nabla \cdot \vec B=0$ and you've now restricted your initial magnetic fields to have zero divergence, so you don't allow just any initial field. In your case if you wanted to try to add this assumption about c, the closest you could get is if you are trying to restrict how you assign clock readings and ruler readings to coordinates to make it so that c will be constant. That's unnecessary if you assign them the regular way, and its too vague if you haven't specified how to connect coordinates to measurements. For instance you allow that people could also have 5d vector spaces in addition to their 4d vector spaces and they could have 5d LT between them and have rulers and clocks and such related to the 5d vector space in the usual way and now c will be constant but this is a different theory with four independent spatial directions. The old 4d vector spaces simply aren't being used to make predictions becasue you never required that have to be.
In case you can't tell, the issue with the LT is that coordinates and LT by themselves don't tell you anything at all. You have to have models that have mathematics and connections between the mathematics and real world stuff. You need to say how coordinates are related to things that can be measured.
The LT don't tell you whether there are things. Or whether they can move inertially. Or whether they can move at c. Or whether they can move at faster than c. The LT by itself is consistent with all of those and with forbidding all of those.
When we do mechanics we'll say things like that the tangent to a worldline of a massive particle always has the same sign for the square. Or we will make force laws and say $F=dp/dt$ and say that just like the massless particles the energy-momentum four vector is tangent to the worldline.
Lots of stuff and connections to the real world have to be made before you get predictions.
