Question: "Did Constantine decide what books belonged in the Bible?"
Answer: It is very important to clarify exactly what role the Emperor Constantine played in the Council of Nicea, what the purpose for the council was, what happened at Nicea, and briefly how the canon—the Bible as we know it—was formed. Constantine was a Roman Emperor who lived from 274 to 337 A.D. He is most famous for becoming the single ruler of the Roman Empire (after deceiving and defeating Licinius, his brother-in-law) and supposedly converting to Christianity. It is debated whether or not Constantine was actually a believer (according to his confessions and understanding of the faith) or just someone trying to use the church and the faith to his own advantage. Constantine called the Council of Nicea—the first general council of the Christian church, 325 A.D.—primarily because he feared that disputes within the church would cause disorder within the empire. The dispute in mind was Arianism, which was the belief that Jesus was a created being. The famous phrase they were disputing was, "There was when He was not." This was in reference to Jesus and was declared heretical by the council and thus resulted in the following words about Christ in the Nicene Creed: "God from true God…from the Father…not made." It was determined by the council that Christ was homoousia, meaning, one substance with the Father.
Concerning manuscripts that were burned at the order of Constantine, there is really no mention of such a thing actually happening at the order of Constantine or at the Council of Nicea. The Arian party's document claiming Christ to be a created being, was abandoned by them because of the strong resistance to it and was torn to shreds in the sight of everyone present at the council. Constantine, and the Council of Nicea, for that matter, had virtually nothing to do with the forming of the canon. It was not even discussed at Nicea. The council that formed an undisputed decision on the canon took place at Carthage in 397, sixty years after Constantine's death. However, long before Constantine, 21 books were acknowledged by all Christians (the 4 Gospels, Acts, 13 Paul, 1 Peter, 1 John, Revelation). There were 10 disputed books (Hebrews, James, 2 Peter, 2-3 John, Jude, Ps-Barnabas, Hermas, Didache, Gospel of Hebrews) and several that most all considered heretical—Gospels of Peter, Thomas, Matthaias, Acts of Andrew, John, etc.
Liberal scholars and fictional authors like to purport the idea that the gospels of Thomas and Peter (and other long-disputed books) contain truths that the church vehemently stomped out, but that simply has no basis historically. It is closer to the truth to say that no serious theologians really cared about these books because they were obviously written by people lying about authorship and had little basis in reality. That is one reason why a council declaring the canon was so late in coming (397 AD), because the books that were trusted and the ones that had been handed down were already widely known.