Let me first say that I do not work in quantum foundations, really, so I might have a few misconceptions myself. I beg anyone to correct me, where I err and I will try to provide more references upon request.
After the question seems to have cleared up in chat, let me rewrite my answer:
You basically seem to ask: What if entanglement would allow superluminal communication in some way. And you propose a protocol. The protocol that you seem to advertise suggests active transfer of information. Since no experiment has ever seen anything like this, I guess we see this as a purely theoretical game.
Let's start with the obvious consequences: If we maintain our usual notions of special relativity, we could suddenly build a quantum cloner (in a way - yes I know there is criticism of this paper, but I don't see how this addresses the problem for us). This would leave us with a big problem, because the no-cloning principle is even more fundamental (in a sense) than superluminal communication, as it derives directly from the linearity of quantum mechanics itself. Already from here, there is no other possibility than to throw aside relativity and reconsider the classical notions of time and space. This is also the usual way I have seen superluminal communication being discussed in quantum information: As a way to show how this would lead to the usual violations of "no-cloning" or "Bell's telephone", etc.
This left aside, let's consider raw entanglement, which you actually want to talk about: Entanglement, as discussed in multiple posts, is nonclassical correlations. Now, when entanglement, as in your protocol, immediately sends information, it is no longer "just correlations", because it becomes active. And I don't see how to get around this "becoming active". At that moment, everything we understand about entanglement breaks apart - because it's no longer correlations, but it would be something completely different: an active link between particles. There are many posts here already describing how this is not possible (e.g. this one).
Now let's consider the protocol again, as the usual protocol involved in Bell's theorem. You claim that "Particle B transmits to particle A, by superluminal signals, the following information", which I interpret as "active transmission of information" - which would violate causality. You claim that this is transmission of information without violating causality and you claim (in chat) that:
All the quantum community researchers, distinguished professors, Nobel prize laureats, try to understand how the entanglement works.
While this is definitely an overstatement ("all" is just too strong), there are certainly many people trying to understand entanglement better. What they try to understand is which types of entanglement can be transformed into other types, how entanglement can provide help in certain protocols, etc. But hardly any one of them tries to understand what entanglement "is". The only answer you need to have in order to work with entanglement: entanglement is quantum-mechanically allowed and classically forbidden correlations. If you don't see
To answer your underlying question: "what is this sending of information with superluminal speed?". First of all, let's fix the terminology: "information", as it is usually defined, is something active: we can do something with it. This is something whose speed is limited by causality. Faster than light speed is nothing unusual: the phase velocity of a wave can be faster than light, even the group velocity of a wave packet can exceed the speed of light (anomalous dispersion case), but in these cases, the speed of information dispersion is still smaller than the speed of light. Since your "information" is something passive, it is no information.
This means that for special relativity, we do not (at all) have to care about something happening faster than light unless it is real information. So I think that your problem is not with superluminal information processing, but with the causal nonlocality of quantum mechanics and therefore, with the quantum state. I believe that you can get rid of the nonlocality by subscribing to an epistemic view instead of an ontic view, but this buys you other problems. This seems at the heart of the EPR-argument: you cannot have a fully ontological (realist) theory and complete locality. However, it is also something that most researchers I know don't care about, because it is too much philosophy for their taste. It is also not necessary, if one only wants a consistent theory to work with.
So I believe that your issue should be addressed in the debate on ontological and epistemological theories for quantum mechanics (see an overview about certain aspects of the discussion). Maybe you will find more satisfying answers there?
Best Answer
Short answer: No.
Short answer as a complete sentence: We have no reason to believe that there is any way that teleportation (or any other quantum mechanical effect, or any other physical phenomenon which we believe occurs) allows you to send superluminal signals.
Detailed answer:
Quantum teleportation can only be used to transfer quantum states. Those quantum states may encode classical information, but they are still quantum states. (Of course, the world being quantum mechanical, even "classical information" is represented by quantum states, albeit those of a very large number of particles at once — but never mind.)
Teleportation requires open classical communication to work anyway; so teleporting the information classically won't save you any work, and in fact is totally unhelpful — except as a quantum mechanical version of a Vernam cipher (i.e. a one-time pad).
Whether anything is secretly happening in quantum mechanics faster than the speed of light is actually a matter of philosophical debate in the foundations of physics. There are people who say that there is (such as advocates of de Broglie–Bohm theory), and people who say that there isn't (advocates of the Many Worlds Interpretation, Consistent Histories, and some Bayesians). What people mostly agree on is that quantum mechanics allows you to realise correlations in probability distributions which are not possible in slower-than-light local hidden variable theories; but that even if there is anything happening faster than the speed of light, you'll not going to be able to use it to transmit signals, because everything looks like correlated but uncontrollable random outcomes.
Related question: Why can't quantum teleportation be used to transport information?