Bible Defended: Moses' Authorship of the Torah

It has come to my attention that some people don't believe that the tradition that Moses wrote the first five books of the Bible is true. Their reason? Well it breaks down to two contentions:

1. The stories were passed down orally and written down later on. Some say during the Babylonian exile. Makes sense? No, this theory does not jive with what the Bible itself says or what the historical record shows. I have two Reasons why I say this.

Had the Torah been codified I think that at least part of it would be written in Aramaic which was the exiles spoke in during captivity. Recall that parts of Ester and Daniel are in Aramaic. However the whole Torah is in Hebrew.

The words of the Torah predates the exile. When the "Books of the Law" were found during the renovations of the first Temple during the time of King Josiah (2 Chronicles 34:14-33). If the Torah was not written down before the Exile, what was they reading?

2. The other contention is that because Deuteronomy describes Moses' death, he could not have written the Torah. There is a Christian response and a Jewish response that I think that works either way. Some Christians believe that someone completed Deuteronomy after Moses died. This does not mean the Torah is not divinely inspired. It's perfect because it came from God. The other reasoning comes from Jewish teaching. They suggest that Moses wrote the passage about his own death in tears. Doesn't bother me because Moses was a prophet and why couldn't he write about his death before he died and be completely right on the details?