Question: I'm interested in finding out:

1. what the process of choosing the books of the New Testament was,
2. when did it occur,
3. who did the choosing, and
4. what are the supporting evidences

Answer: 
Let me give you the original document of Council Of Carthage AD 419 

> 
> Canon 24. (Greek xxvii.)
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> That nothing be read in church besides the Canonical Scripture
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> Item, that besides the Canonical Scriptures nothing be read in church under the name of divine Scripture.
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> But the Canonical Scriptures are as follows:
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> Genesis.  
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>  Exodus.  
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>  Leviticus.  
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>  Numbers.  
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>  Deuteronomy.  
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>  Joshua the Son of Nun.  
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>  The Judges.  
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>  Ruth.  
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>  The Kings, iv. books.  
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>  The Chronicles, ij. books.  
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>  Job.  
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>  The Psalter.  
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>  The Five books of Solomon.  
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>  The Twelve Books of the Prophets.  
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>  Isaiah.  
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>  Jeremiah.  
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>  Ezechiel.  
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>  Daniel.  
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>  Tobit.  
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>  Judith.  
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>  Esther.  
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>  Ezra, ij. books.  
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>  Macchabees, ij. books.  
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>  The New Testament.  
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>  The Gospels, iv. books.  
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>  The Acts of the Apostles, j. book.  
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>  The Epistles of Paul, xiv.  
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>  The Epistles of Peter, the Apostle, ij.  
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>  The Epistles of John the Apostle, iij.  
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>  The Epistles of James the Apostle, j.  
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>  The Epistle of Jude the Apostle, j.  
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>  The Revelation of John, j. book.  
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>  Let this be sent to our brother and fellow bishop, Boniface, and to the other bishops of those parts, that they may confirm this canon, for these are the things which we have received from our fathers to be read in church.
> 
> 
> 

Here, the council take a decision to send the list to BONIFACE bishop of Rome, again in the council of Hippo, it was mentioned that "send the list to Rome and ratify it" It was the Roman Church who decided the canon, based on apostolic tradition..
When the Canon of the Scripture was closed, it already contained a list of books that were widely accepted as authoritative in the Christian communities spread all over the known world.

What we know today as deuterocanonical (or apocryphal) books of the New Testament were considered heretic very early in the history of the Christian communities. So there was never really a given time when people gathered and examined books to decide which ones would be included and which ones wouldn't.

[Wikipedia](http://en.wikipedia.org/wiki/Biblical_canon#Christian_canons) has some interesting information about the Christian Canon.