Logos is primarily connected with speaking; it comes from legein (= speaking = distinguishing, identifying and collecting in language). Logos is the speech and word. It is from this sense that expands to the sense of cause and conceptual explanation, comparison, distinction and identification. You can see it for example in Plato's Phaedo 62, where suicide is said to be unacceptable even when life has become a burden. Translation here betrays the original, as is usual. The original uses the word logos:
"I admit the appearance of inconsistency [alogon = without logos], replied Socrates, but there may not be any real inconsistency after all in this [the original emphasizes the positive side, and, again, uses logos : there must be some logos for this]. There is a doctrine [in the original: logos, in this case a traditional saying, almost a myth] uttered in secret that man is a prisoner who has no right to open the door of his prison and run away; this is a great mystery which I do not quite understand. Yet I, too, believe [that is said well = eu legesthai - totally omitted in the translation] that the gods are our guardians, and that we are a possession of theirs. Do you not agree?" (More)
Then Socrates begins to explain why suicide is unaccepted.
There is, in this small excerpt, all the foundational meaning of logos, as simultaneously cause, purpose, explanation, reasoning, narration, saying, speaking. We must also remember that Plato writes after Herakleitus and his special use of Logos. All of this co-exist in Plato and in the language itself, mirroring a whole tradition belonging to this crucial word, used also by St. John in his Gospel, where Logos is Christ. There is no way of identifying Greek logos with the modern concept of pure reasoning. Logos contains reasoning but is far from limited to this.
Thinking, as we see in Plato and the ancients at large, can only be human when it is based in the whole life, as is expressed by the common meaning of the words logos and legein, the simple speaking that belongs even to everyday 'unimportant' aspects of our life. What we call reason for the ancients can be obtained from many sources. For example, if we say that a cold weather produces sickness, this is true, but without logos. If we know that sickness is produced by a specific influence of cold to our body, which we can describe, then we have logos. Such a knowledge is not always the result of proofs. When Plato proceeds from argument to argument, often takes for granted things that he doesn't prove. E.g. in Sophist 265 he divides art in divine and human, although he admits that many people don't recognise divine art, but believe in an accidental production of beings by nature, and yet he does not prove the existence of God.
There is involved intuition in reasoning, something obvious in the way Plato reasons, and also described explicitly by himself in his autobiographical 7th Epistle. Our admiration for him comes from many things, one of them being precisely this intuition. If Plato was just saying a = b, b = c, a = c, he wouldn't be Plato. He contains this element, but he is not much interested in this. In his speaking many and most important arguments come out as if some revelation has happened. He is a poetical mind. But, (with the exception of Aristotle), ancient philosophy is not what we call by the same name, and it contains poetical thinking and the whole life, especially love life.
This exists in language too, where philosophy comes from philein (to love) and sophia (wisdom). It is a love towards wisdom, but simultaneously and even before, the wisdom that comes out of loving. This can not be recognised in our word for philosophy, because it is not ours, it is an adoption of the Greek word, a cell empty of meaning, a transliteration, and we forgot what this word means and out of which words comes.
I hope this first approach can give us some fuel to continue. But I keep discovering again and again how important language is, how little can we rely on translations.
Reading the Republic, I thought back at what you wrote here. Can we say that logos is also the rational principle according to which the world is made?
Socrates in  says that we "must contemplate [the soul] with the eye of reason, in her original purity."
First, is the word 'reason,' in this case, a translation of logos? Describing how the City and the soul ought to be organized, then, can we say that logos is also the rational principle of the soul, as opposed to the wild and irrational?
When John the Theologian opens his gospel with the words "ἐν ἀρχῇ ἦν ὁ Λόγος," he means thereby that the world was created and is maintained by a non-temporal and immutable, a 'rational', pattern. In order to gain freedom, man must conform to this pattern, the divine logos. Is this also the meaning that Plato intended?
It is remarkable how powerful words are to create. The world was created through the divine Word, according to John, and Plato holds that dialectic is the only science that leads to finding the divine and to freedom.
The word translated to reason, in your republic quote, is not logos but logismos.
Your antithesis between wild and rational reflects a Western foundational tendency of conceiving reasoning as the other-side of an otherwise beastly creature. Since reasoning includes or is essentially an artificial activity (it involves the creation of symbols, administration of concepts through combinations of symbols, etc), the gentle side of man started to look less real, less natural, than man's dark side. Attempts for the other side of reason to be assigned a positive value (movements for returning to "mother nature", nudists, seekers of an original innocence of man which is grounded on sentiment rather than reasoning), never achieved enough power to overcome the other tendency, of man as naturally a beast. You yourself could find a scheme to juxtapose rational and irrational without focusing exclusively to the "wild", you could say, e.g., rational – emotional.
Thus we have in the West the prevalence of a simultaneously overestimated and understimated reasoning!, which, on the one hand is considered higher and worthier, and on the other hand, artificial and unnatural! Therefore, to the degree that he cultivates reasoning the western man is obliged to discover also that in the bitter reality he remains just a wild beast. Out of this comes the (western too) sense of humanistic culture (arts, literature, music, etc) as something external, a polishing or a taming, whence it becomes the last pages of newspapers and magazines, i.e. always less significant, inferior to the most important issues of politics and economy.
According to Plato the world as created exhibits a rational shape. There is a difference between this understanding of ours and the origin of this shape in Creator's mind. God is considered the origin of the Ideas, and Ideas for Plato are not schemes of reasoning but virtues (the Good, the Beautiful, etc). Godly ideas produce the worldly harmony, as a relationship between the elements and forces. The description of such an harmony by means of symbols, the administration of this harmony through symbolical concepts and structures, our reasoning and science, is a second reality, like an artist creates an inferior reality by imitation.
St John uses the word Logos, but since Logos has a thousand meanings, we can not understand its meaning here by just the word itself, let alone infer that it is the Platonic meaning, when even for Plato we don't know if we understand well the meaning(s) of logos in his works.
For St John we are sure at least that logos is not a pattern, it is a person. Logos is Christ. And for Plato we are sure that logos is not a pattern, it is the Goodness inherent in the Creator together with all the other virtues.
"There is a doctrine [in the original: logos, in this case a traditional saying, almost a myth] uttered in secret that man is a prisoner who has no right to open the door of his prison and run away"
I too have heard this myth but from other sources, in Iranian Sufism, in the myth of the heavenly twin (Mandeism and Manicheism) and in the Jewish mystical tradition there are likewise many illusions to this theme as well as in the whole Gnostic tradition (see especially some of the texts of Nag Hamadi). All these sources perhaps (including Plato’s myth) can be traced back to Babylon. Yet, if man is indeed a prisoner why does he not have the natural right to escape from his prison if it has become so unbearable for his soul and spirit to continue enduring? Is not dying while trying to escape not nobler and truly more human then simply accepting an unjust punishment and the consequent suffering for a crime one didn’t know one had committed in the first place?
Yet I, too, believe [that is said well = eu legesthai - totally omitted in the translation] that the gods are our guardians, and that we are a possession of theirs. Do you not agree?"
But who are these Gods? And how do we know they are "our" Gods? If they are indeed our Guardians, is it they who also have us guarded in the prison of this world? I'm not too sure what you mean by us being a possession of theirs, are we a possession in the sense that they own us and, they can simply do what they will with us? If this is true then the notion of us having any freedom is simply yet another carefully crafted illusion that our owners have created in order to "keep us" docile like animals. In such a universe forces beyond our mortal grasp and insight determine every single detail of our lives. In the end we are left asking whether such a life is really a life worth living, since these Gods (could they be Gods?) control every fibre and breath of our lives.
Thinking, as we see in Plato and the ancients at large, can only be human when it is based in the whole life, as is expressed by the common meaning of the words logos and legein
This is very true, yet in the modern world this has become almost impossible to achieve (unless one takes up the life of monastic withdrawal) as modern thinking mostly has devolved into a form of desiccated, third rate abstraction, that can be socially applied and "administered" en-mass to any problem or situation. Such lives are longer whole and this is the way the modern world with it vast ideological systems wants its citizens to be, namely isolated, divided, weak and forgetful of what the good life could of being.
I don't know which type of prisoner is worse, the ones the Gods imprison or the others which humanity makes.
Plato needs a different approach than what we use when we read most of the modern philosophers, if not all. Although he defined argumentative thinking once for ever, he was not confined to it; he uses myths and humor, and many times he speaks symbolically. If we ignore this element and read him as Karl Popper reads him, we arrive at mistakes, and then we start speaking about "totalitarian republics" or gods-tyrants. We must also have in mind all of his works, i.e. the broad context, when we think on a particular work or a passage of a work.
That we don't have the 'right', means that we should not escape, since God's purpose is to heal our wounds, which happens in this life. It seems like a prison, but it is a hospital! As Baudelaire said, we resemble the patients of a hospital who complain about not being next to a window or for other things like this, when they should really understand that they are there not to make a vacation, but to regain their health. For Plato and for the Fathers of Christianity, nothing is a matter of luck; even lying next to a window or not, is part of our treatment. As St. Basil asks the rich man, "is God unjust? Why are you rich and the other is poor, if not for you to learn charity and for the other to learn patience?" (See St. Basil, Where did you find your property?)
That we don't have the 'right', means that we should not escape, since God's purpose is to heal our wounds, which happens in this life I don't know if I can agree with this idea, it's too similar to that rather inadequate, partially enlightened theological theory, that no matter what, some degree (or at least a kind of minimal return) of good must always emerge from the most terrible evils and sufferings that have occurred in the historical continuum of this world. I can, but only reject this rationalised, metaphysical theory as pure nonsensical rubbish, for in fact it actually insults both the suffering done to humanity and the attempt to heal those wounds by appealing to that which makes us truly human, namely our rationality as perhaps Plato understood it.
However, I really don't know if the ancient Greek philosophical tradition, including both the thought of Plato and Aristotle did, and indeed can adequately address the moral and indeed ontological problems that evil posses to the modern mentailty. I believe the problem is particularly difficult for ancient Greek thought to address, because rationality in Plato's time was always understood to be something that was of the highest good. For us, rationality, or what it has perhaps developed into, has and often is directly implicated with violence and evil in the modern world, particularly with regard to technology and its deployment. And it this element of a pure rational technique that has somehow developed in the absence of what was once thought of as being only a part of a whole philosophical tradition in ancient Greece that makes these problems difficult to address now, in our very fractured remains of a once complete corpus of thought. Suffering is sometimes simply best described as suffering and, to imbue it from above with some rational meaning is both dishonest and disingenuous.
Likewise it's difficult to use Plato's notion of "right" nowadays, as the notion of right no longer has any sense of its original moral and metaphysical significance, this again can be traced back to the fracturing process I descried above.
It seems like a prison, but it is a hospital A mental hospital! Why are asylums and prisons the great themes of modernity i.e. Kafka, Beckett, Bulgakov.
As Baudelaire said, we resemble the patients of a hospital who complain about not being next to a window or for other things like this, when they should really understand that they are there not to make a vacation, but to regain their health Health like in the sense Nietzsche sought and proclaimed before he went mad. In a sense is not a great and exceptional mind like Nietzsche's that becomes insane not really a case of a person cutting their life short despite the fact that they may live physically for another decade. Is suicide not an understandable right of appeal to something greater themselves, do not such individuals who effectively "loss it all" (including their body, mind and soul) not have the right to leave early?
As St. Basil asks the rich man, "is God unjust? Why are you rich and the other is poor, if not for you to learn charity and for the other to learn patience? I'm surprised St. Basil got involved in answering such questions, if only heaven were as easy to acquire than a pot of gold!