Your question will be evaluated in two parts:
A) Is Shiite the right religion and does it guarantee prosperity and protection of man?
B) Is truth restricted to Shia religion? If so, is it compatible with divine justice as you have asked in your message?
As for the first part of the question, we must say that in order for a person to be able to learn all the teachings and doctrines of a religion or an entire ideology, he must consider a lot of time. He must reach the truth through studying, asking questions and getting answers, on various subjects and by comparing them with those of other religions. A small number of such arguments can be found in answers coded 2163, 2011, 1252 and 1145 on the same website. For further information, you can refer to books authored by prominent Shia scholars in this regard.
However, considering the nature of your question, it is inferred that you have no doubt as to the first part of the question i.e. the righteousness of the Shia religion and you have doubt, more or less, concerning the second part. We are now going to provide you with some information and details in this regard:
We need to sort out the second part as well into at least four parts so as to come to a better conclusion:
1 – What is justice of God?
2 – Does a religion being right (to the exclusion of other religions) mean that it is against divine justice?
3 – Is eternal happiness confined to Shi'ism?
4 – How does God deal with people who are not Shia?
We will follow the same sequence to deal with the four parts:
1- When it comes to the first part, we must say that divine justice (justice of God) in this respect means that God, the Wise, holds everyone responsible for his deeds and beliefs according to his knowledge and capacity. Naturally, if there are two people who  enjoy completely equal privileges and have the same capacity but one of them follows the right path and the other follows the wrong one, their status will not be equal when it comes to giving reward. Naturally, as stated above, the position of people who reject the truth and do not embrace the right religion knowingly will not be the same as those who do not choose the right religion because they did not get the message of truth. They will not be treated or rewarded equally.  Each one of them should be answerable to God for their deeds in a different way. Kindly consider the following verses so as to understand that such a notion is derived from the Holy Quran:
First: Whoever goes aright, for his own soul does he go aright; and whoever goes astray, to its detriment only does he go astray: nor can the bearer of a burden bear the burden of another, nor do We chastise until We raise a messenger.[1]
Second: And whoever acts hostilely to the Messenger after that guidance has become manifest to him, and follows other than the way of the believers, We will turn him to that to which he has (himself) turned and make him enter hell; and it is an evil resort.[2]
Third: Surely (as for) those who return on their backs after that guidance has become manifest to them, the Shaitan has made it a light matter to them; and He gives them respite.[3]
Fourth: And We do not lay on any soul a burden except to the extent of its ability, and with Us is a book which speaks the truth, and they shall not be dealt with unjustly.[4]
There are many other verses which clearly indicate the general Islamic principle that only those people are held responsible to whom the truth has been revealed and who oppose the truth despite knowing it. Obviously, punishing such people will not be against divine justice. According to us, God will deal with those non-Shiites who for any reason could not embrace the truth differently from those who rejected the truth out of pure grudge and hostility. We will provide further details in the following sections.
2- It goes without saying that two different and sometimes opposing ideas cannot possibly fully represent a single truth. It is like telling someone that two roads going to different directions lead to the same destination! Of course, we should note that a part of every school's teachings and doctrines, even those of non-divine schools, may bear some truth in it.
Therefore, if God says that no religion is superior to any other religion and that every individual can, without investigation and research, follow any religion irrespective of whether it is true or false and that he will attain salvation and happiness with it, such a statement runs counter to divine justice. Because in this case, those who, with their own efforts, follow the right path will be equal and have the same value and credit as those who intentionally and knowingly follow the wrong path. Such a thing is against justice and fairness of Allah.
For this reason, if we have complete faith in the truthfulness of our school, which believe to be more perfect than other schools, we have not uttered a word against divine justice.  When it comes to the duty of those who have not attained the truth, it is a question that has nothing to do with Shia being the right religion. Therefore, it will to be studied separately.
3- Although we believe Shia is the most perfect religion and conscious adherence to it and implementation of its teachings can guarantee man's complete happiness, a study of the teachings of other religions also reveals that some of them are not opposed to Shia teachings. In fact, there are certain principles and teachings which have been accepted almost by all religions. In other words, since such things are imbedded in the God-gifted nature of all human beings, no school can actually deny them.  The belief in the necessity of keeping one's promise and doing acts of kindness to parents, the evilness of oppression, injustice, tyranny, stealing and such other acts can be considered to be among such teachings. It is also necessary to note that Shiite Imams (AS) have taught us that God, the Exalted, has determined two basic criteria to regulate human behavior, one of which being the reason or human intellect gifted to every human being. The other criterion has an exoteric aspect and that is the teachings of Prophets and Imams (AS).[5] Indeed, the greater and sharper one's intellect the more would be his tendency to follow the prophets.  It can be said that the findings of the intellect being separate from his presumptions are congruous with religious teachings.[6] Naturally, men can, with the power of God-gifted intellect, distinguish most of the good and bad things and act according to them even if he may not have a religion, thus attaining partial happiness.
In other words, we believe Shia is the most perfect of religions but we do not believe that he who does not believe in this school will not attain any stages of happiness because one of our beliefs is that the human intellect can independently distinguish many of the good and bad things. As well, there is some truth in the teachings of the divine prophets which have been distorted partially and left imperfect. Non-Shiites or followers of other religions can, to some extent, get close to the right path with the help of those teachings. In the Holy Quran, God has praised some Christians simply because they enjoyed some good traits such as humbleness.[7]  The same good feature will allow them accept and follow Islam, if it is made clear to them.
The conclusion that can be derived from this section is that we cannot say that people who oppose Shia despite knowing that it is the right religion will be deprived of all sorts of happiness at any stage. However, according to us there are people who have failed to follow Shi'ism not because of holding grudge against it or because of arrogance but because they have not been able to get acquainted with this religion. If such people act upon the natural human principle and endeavor to reach the truth, they can have a happy ending. As the fifth Imam of Shiites have said, the criterion for accountability on the Day of Judgment are the intelligence and knowledge with which man has been bestowed in this world.[8] This is a just standard for assessing happiness and prosperity of people. God, the Exalted, shall not want mankind to do anything further as we shall discuss in detail in the fourth section below:
4- The fourth question is: What is the fate of Shiites in the next world and what would be their status there? This is not a new question as it used to be put forth even in the time of the Imams as evidenced by reports and traditions.
A precise and thorough analysis of the verses and traditions will help us separate people into three main groups:
First group:  People who are not Shiite and who trample upon the general natural principles that are acceptable to all human beings simply because they are either ignorant or due to grudge and arrogance. Not only these people have failed to follow the right path or deliberately wanted to stay away from the path of divine leaders but they have also not paid any heed to the internal and esoteric criterion i.e. the human intellect which is yet another bounty of Allah gifted to man. Hence, one cannot hope that they will be protected against the Fire on the Judgment Day.
Second group: People who are aware of Shia being the right religion but grudge, arrogance or fear of losing their position cause them not to follow the right path. Such people, as described by Imam Sajjad (AS), will not benefit from religious deeds even if they keep worshipping Allah for a hundred years while fasting. Their worship will be of no avail to them.[9] The reason is clear and tangible; that is, your religious deeds including prayer will benefit you only when they are for God. Those who know of a truth and of something being right yet they deny it, are surely not seeking to gain the pleasure of Allah but they are in the heart of heart pursuing their own material and worldly ends. The late founder of the Islamic revolution of Iran, Imam Khomeini, writes in his Last Will and Testament after mentioning Hadith Thaqalayn which is one of the traditions proving the rightfulness of the Shia religion: "If there is an excuse on the part of the ignorant ones, there is not any excuse on the part of the scholars of religions."[10] Of course, God's justice in terms of how they will be rewarded for their good deeds in this world would be in such a way that God will keep away some hardships from them in this world such as the pain and agony of death and they will see some of their wishes fulfilled so that they may not be entitled to any rewards on the Day of Resurrection.[11]
Third group: They are a group of people who have not converted to Shia, not because of grudge, arrogance and hatred but because of a lack of proper access to facilities and means to enable them to do the necessary research in this regard. Perhaps, some of them might have done the research but they did not get to the truth. In other words, they are the ones whom we term as "the weak" (mustadh'afin). Most of these people observe, at least in their daily normal life, the natural human principles avoiding injustice and transgression upon others. We must know that most non-Shiites are included in this category and your question also relates to these people.  Your confusion, concerning how God may treat them with injustice subjecting them to hell fire and denying them happiness, is logical and correct.
We must say in response to your question that in the Shia culture, such people are referred to with different names and titles such as "the weak" (mustadh'af), "the dwellers on the Heights" (Ashaab al-A'raf), or/and "those who are hopeful of receiving divine blessings" etc.  According to us, these people do not necessarily enter Hell and the path to happiness is not completely shut to them.  Many narrations have been reported from Infallible Imams (AS) and we are going to mention only two of them:
First narration: Zurarah narrates that he asked Imam Baqir (AS): "Who is a mustadh'af?" The Imam replied that he was a person who neither decided seriously to disbelieve to be called a kafir (unbeliever) nor was the way of faith clear to him. In other words, he neither has the ability to have faith nor the ability to disbelieve. They are a group of children; men and women who have the intellect of children are also included in this category. Such people are relieved of obligation.[12] Elsewhere, the fifth Imam (AS) says to the same narrator: "God can, through His grace and bounty, take the "weak" (mustadha'afin) to Paradise or take them to Hell due to their sins but He will not, however, do injustice to them (giving them rewards proportionate to their deeds).[13]
Second narration: A number of prominent Shiite figures told Imam Baqir (AS) that they liked whoever followed their beliefs and detached themselves from whoever who did not follow their beliefs. The Imam referring to a verse from the Holy Quran criticized their method and said that there were "weakened" (mustadh'af) people among Shiites who rested their hope on God's mercy …[14]
Such traditions have been transmitted repeated by many a number of reports leaving no doubt as to their authenticity. Some Shiites in the time of the Imams (AS) asked them as to what was the difference between Shiites and others if both of them entered Paradise.[15] The answer is that Paradise has different degrees and stations and every individual will go to one of them depending on the nature of his actions. Surely, the station of the knowledgeable and the ignorant will not be equal.[16]
In any case, we must know that God's blessing and mercy is so extensive that, as the Commander of the Faithful, Imam Ali (AS) says in Du'a Kumayl, if God did not decide to ordain Hell for the sinners and a permanent abode for grudging ones, He would also make hell cool and peaceful and would not subject anyone to fire.[17] But since the existence of Hell was necessary for doing justice and God also warned that believers and infidels would not be rewarded in the same way[18], we cannot consider it to be against divine justice and mercifulness.
We pray to Allah to grant you success.

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