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: 
This is hard to reference because it's a fairly organic process, but the works translated by [St. Jerome](http://www.newadvent.org/cathen/08341a.htm) into Latin to make up the Vulgate (People's) Bible are the ones that were read during early liturgies. 

From a doctrinal stand point, this is the other answer:

> 
> The Synod of Hippo, in northern Africa, in 393, approved the New
>  Testament, as it stands today, together with the Septuagint books, as
>  did the Councils of Carthage in 397 and 419. St. Augustine regarded
>  the scriptural canon as closed, as did Pope Damasus I and the Council
>  of Rome in 382. Damasus' commissioning of the Latin Vulgate edition of
>  the Bible, c. 383, was instrumental in the fixation of the canon in
>  the Latin speaking world.
> 
> 
> 

[Source](http://www.rev-know-it-all.com/2010/2010---06-13.html)
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.