To start, I'm going to correct the O.P's quote.
FOR MORE THAN THREE CENTURIES THE CHRISTIAN CHURCH HAD NO NEW TESTAMENT. Not until the Muratoriun [sic] Canon (350 A.D.) did the Christian church begin to compile a New Testament that resembles the one we have today.
There are a few flaws in this conclusion. First of all, the text in all caps is just wrong. The church had the New Testament from the time it was written. Quoting Wiki here,
For the Orthodox, the recognition of these writings [the Canonical Gospels, Acts, letters of the Apostles, and Revelation] as authoritative was formalized in the Second Council of Trullan of 692, although it was nearly universally accepted in the mid 300's.
These writings were not first accepted in the fourth century, they were universally accepted. They were seen as scriptural far before then. Even before the NT was written, the Church had a set of creeds, confessions of faith, and hymns. These can be easily dated back to less than five years after Jesus' death/Resurrection. The most famous of them is written down in 1 Corinthians 15:3-7 (Wiki article). The point here is, the early church had its beliefs set downright from the get-go. The claim that the Church had no unified system of beliefs until the fourth century is rather incorrect.
Now to the meat of the question: Thus can these Gospel faithfully record the actual life of Jesus in the way he lived and the things he claimed and believed? I'm going to focus this answer on these two questions:
Have the gospels been preserved accurately?
Did the original gospels accurately describe what actually happened?
I'm going to analyze the text itself rather than the events here (this is History SE, not Christianity SE). (Fredsbend has written a good answer about other aspects)
Have the gospels been preserved accurately?
The Gospels are commonly thought to be written in this order:
However, this doesn't really work. The Gospel of Luke is addressed to "Theophilus". The Acts of the Apostles (also written by Luke) is also addressed to Theophilus. However, Acts refers to a "former book".
In my former book, Theophilus, I wrote about all that Jesus began to do and to teach Acts 1:1 (NIV)
This is referring to the Gospel of Luke. So, the Gospel of Luke was written before the acts of the Apostles. Now, the Acts of the Apostles ends abruptly with Paul the Apostle (the central figure of the latter half of the book) under house arrest in Rome (A.D. 62)(Acts 28:30). This implies that the book was written before Paul's death. Thus, the latest that the book could have been written was A.D. 62. Now, the Gospel of Luke was written before the Acts of the Apostles and incorporates some parts of the Gospel of Mark. Give a year for each book and the Gospel of Mark was written (at latest) in A.D. 60, possibly even in the fifties. That's less than thirty (possibly twenty) years after Jesus' death (A.D. 30-36). For an event to be recorded that soon after it happened was rather uncommon for that period of history. We don't usually doubt accounts of Alexander the Great, yet the earliest manuscripts mentioning him (the Diodorus) were written in the first century B.C, well over three hundred years later. But here we have an account written in the same generation! Some claim that the average lifespan was about 35 years at that point in history, but that is including infant mortality, which was quite prevalent. The average lifespan once age 15 was reached was almost sixty years. In that little time if the Gospels had included something contrary to what happened, anyone could have gone and refuted it, "I was there and that's not what happened." However, there are no records of anyone disputing the content from that time.
We don't have the original Gospels. The earliest manuscript that we have is a part of the Gospel of John from about A.D. 100-150. So, how do we know that the Gospels weren't changed in those two-three generations? Well, first of all, the church always had a high bar of excellency when it came to copying manuscripts. In the five-thousand plus Greek manuscripts of the NT surviving today, there is only .5% variance. That's 99.5% accurate! Of the slight error, no major doctrines are affected. you can pretty much assume that even less error occurred in that short time-span of less than one-hundred years. Now, the time between the original manuscript and the earliest surviving copy might seem a lot. However compared to other historical documents, it is relatively short. For example, Josephus wrote The Jewish War in about A.D. 75. We have only nine Greek manuscripts, one Latin manuscript, and a few Medieval Russian manuscripts of it. Of these, the earliest was written in the fourth century and the next was written in the tenth century. As far as number of manuscripts surviving, the NT has more surviving manuscripts than any other work of classical antiquity. The runner-up (as you might call it) is the Illiad, with less than 650 Greek manuscripts remaining.
So, this answers the first question, "Have the Gospels been preserved accurately?" Yes, what we have today is the practically the same text that was originally written two millenia ago (give or take a few years :P).
Did the original Gospels accurately describe what actually happened?
This is a large question in itself, but I will try to consolidate my answer to a few main points.
First of all, we have to look at the intentions of the Gospel writers. What were they trying to write when they wrote the Gospels? Let's take a look at the Gospel of Luke.
1 Many have undertaken to draw up an account of the things that have been fulfilled among us, 2 just as they were handed down to us by those who from the first were eyewitnesses and servants of the word. 3 With this in mind, since I myself have carefully investigated everything from the beginning, I too decided to write an orderly account for you, most excellent Theophilus, 4 so that you may know the certainty of the things you have been taught. Luke 1:1-4 (NIV)
You can see that Luke clearly intended to write an accurate account about what had happened. The Gospel of Luke is the only Gospel to have a preface like, that but the styles of the Gospels of Matthew and Mark are quite similar. It is only reasonable to presume that they had similar intents. The Gospel of John is a little different in style, yet we do find a verse in it stating its purpose.
31 But these are written that you may believe that Jesus is the Messiah, the Son of God, and that by believing you may have life in his name. John 20:31 (NIV)
Now, this is more of a theological statement than Luke's, but If you're going to be convinced enough to believe what he says, the theology has to come from accurate history. So, we can conclude from this that the writers of the Gospels did intend to write accurate accounts (think biographies).
Now, they intended to write accurate accounts, but were they able to follow through with their intents? Writing an account of what happened twenty to thirty years later, they could have forgot what actually happened.
However, it is highly unlikely in those (less than) thirty years that the Gospel writers forgot what actually happened. Writing was relatively rare in the Middle Eastern culture where they lived and almost everything was oral. Stories weren't written, they were memorized. In fact, some people would memorize the entire Old Testament (huge). They were able to do so because of this oral culture and being used to memorizing everything. Given this, it's hard to imagine that the Gospel writers forgot how things happened.
A few people have argued that the Gospel writers were dishonest or amoral, but there is no evidence to support this. Rather, on the contrary they are portrayed as people of integrity in the Gospels. Of course, this begs the question, "Did the Gospel writers tell the truth, but paint themselves as good?" A closer reading of the Gospels clearly refutes this, however. The Gospel writers often included details for no apparent reason that were embarrassing or made themselves look bad. For example, Mark has a consistent rather unflattering view of Peter, the head disciple (I'm not advocating the Petrine theory here) (Mark 8:31-33, Mark 14:37, Mark 14:66-72). Other examples include the disciples repeatedly misunderstanding Jesus (Mark 9:2-12), wanting the places at Jesus' right and left hand (Matthew 20:20-24, Mark 10:35-44), and many more. My point is that the disciples told the truth even when it was unflattering.
I've seen a few people using this argument so, here it is. The disciples weren't liars, they just weren't correct either (i.e., they were confused). However, if they were just confused, when they wrote down the Gospels , anyone could have said something like, "hey, I was there too, and that's not how it happened! These Jesus freaks will tell you Jesus worked miracles, but I'm here to say He didn't." There was nothing to keep people from saying this. In fact, the disciples had many enemies who would rather not (understatement) see the Christian religion thrive. They had political and religious agendas that conflicted with it. However, nobody did that and refuted what the Gospel writers said. If they had, Christianity would have died in its infancy. And, surely, the Gospel writers were not the last people alive who had witnessed what had occurred. In fact, in 1 Corinthians 15:6 (commonly thought to be written about 53-54 A.D.) states this.
6 After that, he appeared to more than five hundred of the brothers and sisters at the same time, most of whom are still living, though some have fallen asleep.
If there were five-hundred plus Christians (brothers and sisters), there definitely would have been more people opposed to Christianity living at the time. It's highly unlikely that all of these hundreds (likely thousands) of people would have died in the next ten years leaving only the Gospel writers.
With these evidences, my conclusion is that the Gospels have been kept true to the originals throughout these two millenia and that the original Gospels did accurately record Jesus' life, how He lived, and what He claimed.
P.s. There are many more proofs for the accuracy of the Gospels that I don't have time to go into today, but if anyone would like clarification on a particular point (or perhaps a point I haven't mentioned), feel free to leave a comment and I'll see what I can do.
Josephus The Antiquities
Michael Martin The Case against Christianity
Lee Strobel The Case for Christ
Frederick Kenyon Handbook to the Textual Criticism of the New Testament
Gary Habermas The Verdict of History
John Dominic Crossan The Historical Jesus