So I’m reading this book The History of the Medieval World by Susan Wise Bauer. The book explains a lot about how Christianity became dominant in the Roman Empire and so on. I'll try to answer your questions as I understand:

• When did Christianity appear in Rome?: We havent known exactly when Christianity appear in Rome. The message of Christianity was spread around the Roman Empire by St. Paul who founded Christian churches in Asia Minor and Greece. Eventually, he took his teachings to Rome itself. The origin of Christianity in Rome, like I said above, is not known. But, the history of Christianity in Rome is well documented. The Bible includes an epistle written by Paul the Apostle to the Christians in Rome. The book of Acts records that Paul, though Jewish, was a Roman citizen by birth.

• Why is Chritianity the religion of the slaves and peasants back then?: At the beginning, the Church was scattered throughout the Roman Empire. Because there was so many cities scattered throughout the empire, Christianity spread within the urban centers, which were populated by slaves and the poorer members of society. Christianity soon became known as "a religion of slaves."

• When was it outlawed or When were Chritians persecuted?: It happened between the Great Fire of Rome in 64 AD under Nero Caesar and the Edict of Milan in 313 AD, this means the persecuting period spanned over 2 centuries. Starting in 250, empire-wide persecution took place by tdecree of the emperor Decius.

• Why was it outlawed: Here is an idea: Social reason: In his book Christianity Through the Centuries: A History of the Christian Church, Chapter 7: Christ or Ceasar, Earle E. Cairns wrote that:

The exclusive sovereignty of Christ clashed with Caesar's claims to his own exclusive sovereignty. The Roman empire practiced religious syncretism (the combining of different beliefs) and did not demand loyalty to one god, but they demanded preeminent loyalty to the state, and this was expected to be demonstrated through the practices of the state religion with numerous feast and festival days throughout the year.^1 However, the nature of Christian monotheism prevented Christians from participating in anything involving 'other gods', so they didn’t participate in feast days or processionals or offer sacrifices to the gods; this caused hostility. And what could make it worse? They refused to offer and burn incense to the Roman emperor, and in the mind of other people, Christians seemed to be disloyal to both Emperor as a god and as the embodiment of the Roman empire.^2 In Rome, "religion could be tolerated only as long as it contributed to the stability of the state" which would "brook no rival for the allegiance of its subjects. The state was the highest good in a union of state and religion."^1 In Christian monotheism the state was not the highest good. Also, back in the day, Christians would gather at night and in secret. They moved the religious activities from the streets to houses. This aroused suspicion among the pagan population accustomed to religion as a public event; rumors had it that Christians committed flagitia, scelera, and maleficia— "outrageous crimes", "wickedness", and "evil deeds", specifically, cannibalism and incest (referred to as "Thyestian banquets" and "Oedipodean intercourse")— due to their rumored practices of eating the "blood and body" of Christ and referring to each other as "brothers" and "sisters".

• What was it like when the first Christians were going around?: During the first century, Roman authorities used the Christians who resided in their city as political scapegoats. Christians received the blame, and punishment, for everything from plagues and economic inflation to hostile invasions by barbarians. Under the Emperor Nero, the public execution of Christians became "sports" events in which the early believers were torched, fastened to crosses and torn to pieces by dogs. In AD 64, part of Rome was burned down. The Emperor Nero blamed the Christians and the people turned on them, followed by arrests and executions. Senator and Historian of the Roman Empire, Tacitus, wrote:

Nero punished a race of men who were hated for their evil practices. These men were called Christians. He got a number of people to confess. On their evidence a number of Christians were convicted and put to death with dreadful cruelty. Some were covered with the skins of wild beasts and left to be eaten by dogs. Others were nailed to the cross. Many were burned alive and set on fire to serve as torches at night. But is was not until the mid-third century that emperors initiated intensive persecutions. And it was caused by Emperor Decius. Persecution during this period was called The Decian Persecution. It resulted from an edict in 250 by Decius ordering everyone in the Roman Empire to perform a sacrifice to the Roman gods and the well-being of the Emperor. The edict ordered that the sacrifices be performed in the presence of a Roman magistrate, and a signed and witnessed certificate be issued to that effect. It was the first time that Christians had faced legislation forcing them to choose between their religious beliefs and death. A number of Christians were put to death for refusing to perform the sacrifices, many others apostatized and performed the ceremonies, and others went into hiding.

• How did it become the main religion of the Roman Empire?: To the main question: It all began when Constantine the Great decided to convert to Christianity. As we all know, the Roman Empire was a big one, and the power is heavily divided, many wars took place. I could name a few: Battle of Issus, wars against the Goths, Carausian Revolt and one of the most prominent: Civil wars of the Tetrarchy. This war ended with the Battle of the Milvian Bridge. Constantine the Great had a vision of the cross appearing over the sun at the Ponte Milvio with the words "in this sign, conquer." His forces defeated and killed Maxentius, which was the sitting emperor of Rome. On the morning of October 29, 312, Constantine marched into Rome “with the sign of Christ marked on each shield”. He defeated the sitting emperor of Rome then which was Maxentius. However, his victory over Maxentius didn’t give him the whole empire. Thirty years earlier, his predecessor, Diocletian, had appointed co-rulers to share the job of running the vast Roman territories which had spawned multiple lines of succession. Two other men currently held parts of the empire. Licinius claimed the title of imperator over the central part of the empire, east of the province Pannonia and west of the Black Sea and Maximinus Daia ruled the eastern territories, which were constantly threatened by the aggressive Persian empire. Licinius didn’t want to war with Constantine so he held his army under the Christian banner and defeated Maniminus (he actually killed himself with poison). Anyway, there were only two rulers left and Licinius allied with Constantine by marrying his half-sister Constantia. The two men met in Mediolanum (modern Milan) to celebrate Licinius’s marriage to Constantia and to issue an empire-wide proclamation that made Christianity legal, which was highly necessary given that both men had now wrapped themselves in the flag of God in order to claim the right to rule. This proclamation was called the Edict of Milan. “No one whatsoever should be denied the opportunity to give his heart to the observance of the Christian religion,” the Edict announced. “Any one of these who wishes to observe Christian religion may do so freely and openly, without molestation…. [We] have also conceded to other religions the right of open and free observance of their worship for the sake of the peace of our times, that each one may have the free opportunity to worship as he pleases.” Property which had previously been confiscated from Christians was supposed to be returned. All Christian churches were to be turned over to Christian control. “Let this be done,” the Edict concluded, “so that, as we have said above, Divine favor towards us, which, under the most important circumstances we have already experienced, may, for all time, preserve and prosper our successes together with the good of the state.”

• Why did Constantine do all of this thing to legalize and spread Christianity tho? Because he wanted to unite the whole Roman empire, he saw in Christianity a new and fascinating way of understanding the world, and in Christians a model of what Roman citizens might be, bound together by a loyalty that transcended but did not destroy their own local allegiances. Christianity could be held side by side with other identities. It was almost impossible to be thoroughly Roman and also be a Visigoth, or to be wholeheartedly Roman and African. But a Christian could be a Greek or a Latin, a slave or a free man, a Jew or a Gentile. Christianity had begun as a religion with no political homeland to claim as its own, which meant that it could be adopted with ease by an empire that swallowed homelands as a matter of course. By transforming the Roman empire into a Christian empire, Constantine could unify the splintering empire in the name of Christ, a name that might succeed where the names of Caesar and Augustus had failed.

So that was how Christianity became the religion of the Roman empire. The rest of the how it became it is today, I think you should read for yourself in the book I mentioned above.

Reference:

1: (Casson, Lionel (1998). "Chapter 7 'Christ or Caesar'". Everyday Life in Ancient Rome; Lee, A.Doug (2016). Pagans and Christians in Late Antiquity: A Sourcebook)

2: A Short History of the Early Church, By Harry R. Boer