When was the New Testament written, and by who?
The NT is the collection of Christian scripture that was held by early Christians since the end of the 1st century, mainly the 4 Gospels and the messages. They are a Koine Greek literature work about Jesus of Nazareth.
These books were not 1-pack, the first time in history when the 4 Gospels were put together was a compilation called the Diatessaron, which was compiled between 160-170 CE.
Before that, the early Christians had "Gospels" and sayings [i.e. oral tradition] about Jesus that started to flow among people as per the testimony of the author of Luke:
Luke 1:2 Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word;
The biblical stand on the authorship of the Gospels is in conflict with the church tradition, the consensus among scholars is the authors are anonymous, because:
- There's no testimony on their identity writing the books.
- The authors themselves didn't want to reveal whom they are, e.g. the ending of John.
- The works are written by educated people who came from Greek communities, rare to find in the old world.
So the scholars [instead] began theorizing on these writings by comparing the Gospels verses together, and also comparing the messages attributed to a specific author to find differences in the way of writing and common belief points, this is called textual criticism.
The dominant theory on the Gospels sources is the Q document:

Scholars assume from the similarities between the verses of Matthew and Luke; that their authors must have been reading from a source, and found that Mark being the shortest; makes it the oldest Gospel.
It's probable that Matthew and Luke authors were reading from the Gospel of Mark to write identical verses, but the common verses [between both] that was not found in Mark were assumed to have came from an unknown written source, and they called it Q.
The 4th Gospel was taken out of this equation because the author of John doesn't seem to share the same theological belief about Jesus like the authors of the Synoptic Gospels [i.e. the first 3], so it must be the last Gospel to be written, and at a later time from them.
Of course, others theories exist on the writing of the Gospels we have today, but the Q document theory is the most accepted one.
Being Muslim myself, I personally have no objection to most of what's written in the Gospels [and maybe the New Testament] even when these books may be under the shadow of faith than historical accuracy. For preservational purposes; they tell us what the early Christians said about Jesus, the man who quarreled with the Roman authorities 2000 years ago. They don't need to be 100% accurate, but at least we can grasp what Jesus taught, and most importantly, the God of Abraham.