Do we know the words used by the original authors of the New
The answer is a qualified yes.
As tniles09 comments, the (scientific) discipline that studies this is called Textual Criticism, and has had an enourmous development, especially during the last two centuries. You can check the current consensus of the scholars in the majority of serious books on the subject (you can discard the obviously "interested" ones, be on the christian apologetic side, in the atheist side, or in the mere sensationalist-commercial side).
To summarize, and restricting to the four gospels (main part of the New Testament), we can state with high degree of confidence:
- The four gospels were written in Greek, in the date range 60-100 AD
(first Marc, then Matthew and Luke, last John)
- The oldest manuscripts we have now are small fragments, the earliest complete manuscripts of the gospels are from the third or fourth centuries. They are, relative to other antique texts, abundant.
- The "original" texts can recovered with high degree of confidence (only some words or verses are in doubt). Errors in copies (both unintentional or intentional interpolations) are easily identifiable, in general. The reconstructed texts essentially coincide (translations aside) with the "official/vulgar" texts of the Bibles used always by the Christian churches.
I quote "original", because your assertion "that is, we know what the original authors wrote" needs some qualification. We need to relativize our modern ideas of literary autorship and originality (the individual that sits alone in his study, and writes "his" own words). Twenty centuries ago, things were different, in many senses: oral tradition played an important part (not only as a trasmissions of facts/tales/ideas, but as a trasmission of texts), the tasks of literary creation (and reelaboration, conservation, trasmission) was more a community thing. Think for example of the "synoptic problem": parts of the first three gospels seem like copy-paste from others, and yet they differ (even sometimes contradict) with each other, and each one has its own style: this would be almost incomprehensible in our modern frame of "autorship" (either we have four individual and honest authors... or we have not; if they are narrators of the life of Jesus, we expect them to tell us the same things using different words; the gospels, instead, sometimes seem to be using the same words to tell different things).
We don't need to go to the extreme, and reject the notion of individual authors: textual criticism asserts that, in spite of the above, each one of the gospels has an individual author, in the literary sense. But this must be weighted against the other elements: they were antique authors, they regarded themselves as faithful voices of a community. Granted this, then yes, we can be reasonably sure that when we read the current texts of the New Testament, we read "what they wrote".
We are not here saying that
- the original authors were indeed named Matthew, Mark, Luke, John - or that they are some identifiable persons (that Mt. and J. were disciples, for example). We don't know that, we know practically nothing about the concrete authors, and we know that is was an antique custom to use pseudonyms to enhance a book authority. (What we know is that Luke was the same author of the Acts book)
- the events narrated are strictly historical, in the modern sense of the word
- the authors were (or claimed to be) "inspired by God" or that the texts are somehow "inerrant" (those are theological questions)