I don't want to discourage further answers but while you wait there's been a few good threads on this in the past that might be of interest including:

Assuming an atheistic perspective, what factors gave rise to the widespread adoption of Christianity in ancient Rome prior to it being made the official religion of the empire? by /u/Zosim

How did Christianity go from a small, hated cult to the state religion of the Roman Empire? What made it so different than the hundreds of other small cults throughout the Empire? ,How significant was Constantine's conversion to Christianity? and Why did Christianity, and not another contemporaneous cult, spread so rapidly within the Roman Empire? by /u/talondearg

How did Constantine's adoption of Christianity change it? by a deleted user

Was there much resistance by Roman citizens to keep worshipping Roman deities in the post-Constantine Rome? by another deleted user

How late did Greco-Roman polytheism / paganism last as a religion after the Roman Empire became officially Christian, and where were the last areas it was practiced? which has a lot of good answers.

Why didn't roman aristocrats overthrow an emperor whose suddenly converted to a religion they'd persecuted for over two centuries?

It sounds like you buy into this narrative of almost constant, organized persecution of Christians by the Roman authorities from the first century on. It's a very popular narrative, but it's a bit overblown. There certainly was persecution of Christians early on; Ignatius of Antioch was on his way to be executed as he wrote his extant letters at the end of the first century, and Christians wrote and preserved hagiographic narratives of martyrdoms, like the Martyrdom of Polycarp, who died around the mid to late second century.

But the reality is that persecution of early Christians was infrequent and regional—not constant and empire-wide. It wasn't as though the whole empire wanted Christians dead. This shouldn't be surprising, as there were very few Christians running around early on. In his history of the early church, Wilken cites an estimate that there were fewer than 10,000 Christians in the whole empire by the end of the first century; that's 0.0017%. Why persecute a group you're hardly even aware of?

But the first general persecution of Christians empire-wide wasn't until 250 under Decius, who ordered all citizens to offer sacrifice to the gods. By this time, Christianity was growing faster—Wilken estimates over a million by 250—and their absence from sacrifice was striking, resulting in the arrest of a number of bishops.

Decius was emperor for only a very short time, and the persecution of Christians under his policy ended with him. But we see empire-wide persecution come again under Diocletian and Galerius in the late third century, who once again required sacrifice to the gods, targeting Christians who refused.

As Wilken notes, this wasn't a particularly successful policy.

But the persecution could not be sustained. Christians were too numerous, their communities too cohesive and organized, and their leaders too adroit to be done in by the sword. By forcing a choice between Rome and Christianity, the emperors badly misjudged the strength and resiliency of the Church. There was little popular support for the policy, and some citizens actually shielded Christians. Diocletian and Galerius represented a narrow and hard paganism that was divorced from the social fact that Christians were not outsiders but neighbors, friends, members of one's family, shopkeepers from whom one bought bread or vegetables or fish. [p. 78]

All that to say: empire-wide persecution of Christians was not the norm before Constantine, and as Christianity rapidly grew (Wilken estimates 10% of Romans were Christian by 300), it became an untenable policy.

You ask:

And last of all, if the majority of the Roman Empire was converted at the time, how did Christians covert such a massive and powerful empire? As far as I know, there was no organized church at the time, while the Greek-Roman pantheon was organized.

As noted above in Wilken, there most certainly was an organized church by this time. There was an established clerical hierarchy of presbyters and bishops. They were organized enough that they needed to have considerable debates about what to do with Christians who lapsed and sacrificed to the pagan gods under Decius, and they were organized enough to withstand persecution under Diocletian and Galerius and prevail.

This demonstrated resilience probably had something to do with Constantine's conversion in 312. Although his historian Eusebius tells us that Constantine had a supernatural vision of the cross, his conversion was likely a strictly political maneuver; religion was inherently political in the Roman empire. One might suppose that, judging by the failure of previous efforts to suppress Christianity, Constantine concluded that Christians could be more useful than the religious traditionalists in advancing his own goals.

Then once the emperor converts, it becomes politically advantageous for others to convert as well.

But that wasn't the end of the matter. There was renewed persecution of Christians under Julian. This excellent comment by a deleted user is all about that.

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I relied on Robert Louis Wilken, The First Thousand Years: A Global History of Christianity throughout this comment.