When was Deuteronomy written?

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I am interested in the theories on when Deuteronomy was composed, and how it relates chronologically to Genesis, Exodus, Leviticus, and Numbers.

As discussed elsewhere on this site, some consider Deuteronomy the work of a later author (or later authors) than the first four books; others consider all five books essentially the work of one author, in roughly the same time period.

I understand that there are, broadly speaking, 4 principal categories of theories:

1. Moses wrote all 5 books (except the last 8 verses of Deuteronomy) circa 14th-century BC

2. All five books of the Torah were compiled from earlier sources (oral & written) principally derived from Moses, but they did not reach their final form until the time of the Babylonian captivity

3. The first four books were compiled from Moses’ work after Moses’ death, and someone else wrote Deuteronomy, using Moses’ books as a source, centuries later

4. None of the first 5 books are based on the work of Moses; Deuteronomy was written last, but all of them were written centuries later

Questions

1. Which of these theories (or a different theory) is best supported by the evidence?

2. Was Deuteronomy written in the same era as the first four books, or is it a much later work?

(see this related question: When was the Torah written or compiled?, which asks about the Torah as a whole, but does not really evaluate Deuteronomy’s relationship with the first four books)