The man was a conqueror winning numerous battles including the famous battle of Milvian Bridge in which he had a vision of the cross before the battle. His men wore the cross on their shields to victory.
So, a bit of debate here, actually. Eusebius mentions his vision of a cross, and Lactantius a 'sign' that may be a Staurogram or a Chi-Rho in a dream. These aren't the same, and though Constantine's Labarum standard certainly was a Chi-Rho later, it may not have been initially, but rather a Latin cross with a curved end - an obvious form to develop from a standard legion Vexilla. A Pagan source, the 7th Panegyric, mentions a vision of three Xs for a 30 year reign. These may be separate visions, but most modern historians think they are the same (usually connected to a solar halo). Initially it was not connected with Christianity at all by other observers, in fact.
1)Was it a good thing for the church that it gained control of the government?
The Church certainly did not gain control of the government. Constantine was a Roman Emperor, and you don't get to become sole Emperor and overthrowing the Tetrarchy, nor remain Emperor, without much blood and strife and effort. Pagans remained in high office (Augustine dedicated City of God to a Pagan Proconsul of Africa for example), as the strife with the mostly Pagan Senate later testifies, and while patronage of State Cults diminished under the Flavians, Theodosius first actively ended them. Even then, Paganism was not outlawed, as even in the heart of Rome the Lupercalia continued to be done into the 6th century!
The Church perhaps helped buttress rule in the latter Empire, but it certainly never took control. Even in the Eastern Empire in Byzantine times, matters of State took precedence - in fact, Justinian would codify a form of Caesaropapism later.
2) Was Constantine a real Christian?
Christianity was certainly a rising idea at the time, so Constantine using it to further his ends as Emperor is obvious. Was it all show? I don't think so. He could have just made syncretic noises with his previous deity of Sol Invictus (where Jesus anyway became often depicted with solar attributes at the time), playing both sides, as others will do in future (like Mongol Khans or Toyotomi Hideyoshi). To expressly favour Christianity certainly cost him some political capital too, as is clear in his later war with Licinius, the building of Churches, and the fallout of his killing his wife Fausta for adultery. His baptism only later in life was normal practice at the time, anyway.
He was a deeply flawed sinner, as are we all. Being an Emperor, his flaws obviously were magnified by the resources at his command. I see no reason to doubt a sincere commitment, which is anyway what the sources claim. He will face God in judgement like the rest of us, anyway.
3) What was the real fruit, in Christian terms, of his reign?
Constantine facilitated processes that were anyway ongoing, such as the Church defining Orthodoxy and such, and speeded up the ongoing conversion of the Empire. The spread of Christianity within it, was remarkable before this, especially in light of official condemnation. It is estimated that lower Egypt was already mostly Christian by the late 3rd century, so in Constantine's day, Christians were probably about 10% or so of the population - nearing the point where exponential growth would have been expected.
The point above of increased laxity makes sense, as far more fair-weather Christians obviously came to be, and the dangers of worldly advancement crept into a Church career, where previously it mostly just offered martyrdom. Constantine was an important figure in Christian history, but secularists and conspiracy theorists like to overemphasise him to a ludicrous degree.
