The answer to the Constantine question is a flat no. All the books of the New Testament were written well before Constantine was even born.
As to the authorship traditions. There was really a Paul and seven of the Epistles attributed to him are universally accepted as authentic. The others are disputed, and most critical scholars do not accept them as authentic. The other Apostolic Epistles (Peter, John, Jude James) are believed to be 2nd Century pseudepigraphical works.
The authorship traditions of the Gospels are late 2nd Century attributions to originally anonymous works. Most critical scholars do not accept the traditions as authentic. The authors never identify themselves but for a wide variety of reasons, including dating language and interdependence critical scholars do not believe they could have been written by witnesses or anyone who knew witnesses. None of the authors even claim to be witnesses.
Conservative Christian scholars have been more reluctant to accept the conclusions of critical scholars (some are obligated by their institutions to only support an inerrantist position), but the mainstream view is as I stated. The Gospels were written between 70-100 CE by anonymous, Greek educated non-witnesses living outside of Palestine. Seven of Paul's letters (Galatians, Romans, 1 and 2 Corinthians, Philippians, Philemon, 1 Thessalonians) are undisputed as authentic. Revelation dates to the late 90's and while the author calls himself John, it is not the same author of gJohn or the Epistles of John. Hebrews, which is sometimes traditionally attributed to Paul, is formally anonymous (the author never claims to be Paul, so it's not pseudepigraphal), but because it seems to talk about the Temple as still being active, it may have been written before the Jewish Revolt, or at least before 70. The rest of the Epistles are 2nd Century forgeries. None of the books of the New Testament are dated any later than the 2nd Century, though.
Bruce Metzger and Bart Ehrman The Text of the New Testament: Its Transmission, Corruption, and Restoration
Bart D. Ehrman The New Testament: A Historical Introduction
Bart D Ehrman Forgery and Counterforgery in the Ancient World