I’m afraid you won’t find this project very satisfying. The PLA is not a doctrine or a set of beliefs. Many people have tried to reformulate it in that way, with mixed success. But it is more than a theory or doctrine; it is a re-interpretation of our language and thinking about pain and, by extension, other experiences. The benefit is that we can escape from various philosophical quagmires.
Perhaps a comparison will help. You may know the story of Dr. Johnson’s “refutation” of Berkeley’s arguments for immaterialism. In case you don’t, Boswell tells it as follows:-
After we came out of the church, we stood talking for some time together of Bishop Berkeley's ingenious sophistry to prove the non-existence of matter, and that every thing in the universe is merely ideal. I observed, that though we are satisfied his doctrine is untrue, it is impossible to refute it. I shall never forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it,– "I refute it thus."
See George Berkeley - Wikipedia.
The odd thing about this is that although Dr. Johnson thought that Berkeley was denying plain common sense, Berkeley is at great pains (both in the Treatise and in the Dialogues) to insist that he is not contradicting common sense; his aim is just to correct a mistaken philosophical theory. In the same way, Wittgenstein’s argument does not try to contradict ordinary non-philosophical life or language, but to re-orient philosophy so that it no longer wanders through a maze of puzzlement generated by a misleading model of introspection.
You’re asking a lot in your list of propositions. In this context, very few of them have straightforward answers. A proper answer to them would disentangle the philosophical models from the everyday use of language, but that would inflate this answer to inordinate length and I’m concerned that it would become a TL;DR or lead to a long and confusing discussion.
At the heart of it is the conventional philosophical doctrine about introspection. This observes that if I say I am afraid or feeling pain or thinking of a beetle, no-one else can meaningfully contradict me. This is what “privacy” means in this context, and it is quite different from the caterpillars and beetles that children used to keep in match-boxes or from some secret vice indulged in only when on one’s own (“in privacy”). Those privacies can be violated. But it is not logically possible to violate this kind of privacy.
So introspective knowledge becomes a gold standard, alongside logic, of what knowledge (certainty) is. The catch is that, just because it cannot be wrong, it isn’t really meaningful to speak of it being right. (The same point led to logical truths being classified as “trivial”).
The argument is that this model of introspection needs to be replaced, and his idea is that “I am in pain” should not be compared to “The patient is in pain”, but to “Ouch!”. It is an expression of pain, not an observation about it, and belongs alongside wincing, clutching the affected part &c. Consequently it is not true or false in the way that ordinary observations are.
The crucial point here is that the symptoms of pain are part of the concept of pain and not an add-on – and, incidentally, so is the impulse to help on the part of other people. (He doesn’t comment on sadism.)
Of course, there is more to be said here. For a start, “I am in pain” is not simply a variant of “Ouch”, but plays a part in the public language game. However, there is no point in my trying to duplicate the various expositions of the argument, which I’m sure you are quite capable of looking up on your own. But perhaps identifying the starting-point of the argument and its terminus will be helpful.
In response to your request, here is my response to each of your questions.
People have private mental lives. Yes, but they can be revealed and/or shared.
People have subjective experiences. People do experience things. Experiences are subjective by definition.
Pain is real. Fake pain and mimicked pain are not real. Phantom pains, psychosomatic pains and referred pains are real. Normally, physical injuries are really painful.
People have P-consciousness. I’m not familiar with this concept; I must be out of date. No comment.
People have qualia. Another concept that I’m not very familiar with. No comment.
A person may go off alone and think by himself. Yes.
Language is used to transmit concepts from one person's mind to another. That is: there was a private concept in one person's mind, then a public utterance, and then a similar private concept appeared in a second person's mind. I accept that one of the functions of language is communication between people. But the question assumes a model of language that I find very problematic.
A person, alone, may use language to conjure images in his own mind. Assuming he does not have aphantasia. (If you are a PLA believer, do you have aphantasia?) Certainly people imagine things and “using language” (using language to do what?) may be a helpful stimulus.
There is an objective reality, independent of what people say about it. Whether my car is still parked where I left it does not depend on what I want or believe or suspect.
A social consensus, even a unanimous one, can be incorrect. Yes, and another social consensus (such as driving on the left or right hand side of the road) can be neither correct nor incorrect.
People often have an internal dialogue. I would have said yes. However, the model that you link to is not wrong, but is extremely problematic.
A person may decide that the word "lurrim" (for example) will be his private synonym for "table," tell this to no one else, and use this word in a meaningful way in his internal dialogue. This is just a variant of question 13. The question is what “meaningful” means.
If a person has a literal, physical, beetle in a box (as Wittgenstein famously discussed as a metaphor), shown to no-one else, then statements the person makes about the literal beetle are still meaningful and may be true or false depending on how well they match the physical characteristics of the beetle. This is just a variant of question 12. The question is what “meaningful” means.
A person may hear what others say, but then he must decide for himself whether he will believe it, based on his own sense of reason. Ultimately, this personal sense of reason is the "gatekeeper" of what beliefs go into a person's mind. Certainly, nobody else can make up my mind for me. But I don’t know what you mean by “a personal sense of reason”. I suspect that a philosophical thesis that I might disagree with is buried in it.
Psychology is a legitimate field of study. Sometimes it is. But it depends on what you mean by “psychology” and “legitimate”. In this respect, it is no different from physics or etmylogy.
Language is primarily a social activity. Surely that’s a cliché. Certainly, it’s not unique to PLA believers.
It can be difficult, often impossible, for a person to accurately share his private mental experiences with others. Yes. But how is it decided when a person has accurately shared his private mental experiences?
As a practical matter, because of the difficulty of sharing private mental experiences, we should avoid speaking about them in a philosophical context. The potential for confusion and talking past each other is too high. That seems like an unwise and unrealistic policy to me. The difficulties can be overcome.
It is better to "think with other people," trying to build consensus and adopting existing consensus, than to ruminate or meditate by oneself. Both solitary and social thinking are to be recommended.