Regarding the Old Testament apocrypha, my understanding is that they were included in the Septuagint (a greek translation of the OT). The translation was done between the 3rd and 2nd century BCE. They weren't part of the Massoretic text and may not have been in previous OT writings.
Concerning the New Testament, I believe there was a great deal of general agreement well before Constantine. The Muratorian fragment, which could date from about 170 AD, mentions the four gospels, Acts, the Pauline works, Jude, and other works.
Several church fathers quote extensively from a variety of NT texts.
I doubt very much that the NT was "chosen by one man" as you say. There was no universal agreement for some time after the first century but by his time the canon was more or less established, IMO.
MIke