At that time, the only written word that was the Word of God was the Old Testament, which Jesus quoted frequently, and even used to show others that He was, indeed, the Messiah that was foretold.
Jesus never wrote a single word of God down.
Not that we know of, true.
Jesus taught. That's what was important to him, not books or words.
Yes, Jesus taught, but also viewed the written word as important, as has been shown.
Yes, but the written word of the Torah. This thread is about the New Testament (or even the Bible as a unit) and Jesus didn't seem to care much about either.
Also, We have John being told to write, by Jesus, Himself:
Rev 1:19 Write the things which thou hast seen, and the things which are, and the things which shall be hereafter;