I am curious whether Jews view Scripture (i.e. the Tanach) as "the word of God," and if so, in what way? Is it "inspired" by God? Is it a "product of the Spirit of God"? Is God the "Author"? [etc.]
In other words, what was God's role in the production of Scripture from a Jewish perspective?
The traditional view:
The Jewish Scripture, i.e. Tanakh, is made of 3 parts.
The first part is the "Chumash", the five books of Moses. They were dictated word-by-word from G-d, and Moses wrote them down. (Now most of Deuteronomy is a big speech of Moses, but even so, after the fact that's what he was ordered to transcribe.) The last few verses describe Moses' death; either G-d dictated them to Joshua (Moses' successor), or Moses had to write them in advance (which must not have been fun).
Just because they are the word of G-d does not mean they were intended literally; some legal material was left unclear and was accompanied by an oral interpretation, and some of the stories involving angels may have occurred during dreams or prophetic experiences. That doesn't make it any less divine.
The Prophets (Joshua, Judges, Isaiah, Jeremiah, etc.) had a lower level of prophecy than Moses. Often it would involve a vision that they had to interpret; the Talmud observes that two prophets could have the same prophecy and while the overall message they would convey would be the same, the choice of words they would use would be different. Thus we would view the words of Isaiah as sacred and the message as being from G-d, but not the same word-for-word "authorship" compared to the first five books.
The other Writings (Psalms, Proverbs, Song of Songs, etc.) were the product of divine inspiration, but not above the threshold of what we'd call "prophecy." (~2000 years ago it was even debated whether Ecclesiastes was truly divinely inspired or simply a great work of wisdom. It got the upvote in the end and it's included in Tanakh!)
Rambam's Eighth Principle (translation by Aish.com):
We believe that the entire Torah in our possession was given [to us] by the Almighty through Moshe Rabbeinu, by means of the medium we metaphorically call "speech." No one knows the real nature of this communication except Moshe, to whom it was transmitted. He was like a scribe receiving dictation. He wrote the history, the stories, and the commandments. Therefore he is called "[the] inscriber."
With regards to the last 8 verses, see here.
