Doug Mason:
In which year did the Returnees first meet at the temple site?
Iyyar (May) 537, as already indicated.
1. If the writer was using a Nisan calendar and accession-year reckoning, Cyrus’ first year began on 24 March 538 BCE.
Correct. Well, close enough. 23 March 538.
2. If the writer was using a Tishri calendar and accession-year reckoning, Cyrus’ first year began on 17 September 538 BCE.
Red herring. The Babylonians didn't use a Tishri calendar, nor did the Bible writers during or after the exile. The Jews also adopted the Babylonian names of the months.
3. Did a first year of Darius precede the rule by Cyrus?
Darius was 'king' (governor) for only a few weeks. Cyrus entered Babylon and took the throne not long after Darius. Darius doesn't have an accession period because he was a governor for Cyrus, not technically a king. (Just as Belshazzar was not actually king despite Daniel's description.)
4. What was the date when Cyrus made his decree?
Red herring. I have already indicated a period of time during which the decree was made.
5. How long did it take for thousands of people to prepare their families and belongings?
The 'thousands of people' could prepare at the same time. It is is not as though each family had to prepare one after the other. Beyond a very small number of people, a very large number of people would not take significantly longer to prepare than a large number of people.
Leaving Babylon was not mandatory. Not all of the Jews left Babylon.
6. What was the date when they started their trek?
Red herring. Most likely about four months before they arrived.
7. Exactly how long did it take before each family arrived in its town or village?
Red herring.
8. Exactly how long did it take for all the families to be settled in their towns and villages?
Red herring.
9. How long did it take for the last one to walk from their home to Jerusalem?
Red herring. Irrelevant question based on a false premise of magical thinking.
10. Where does the Bible say that this event concluded the “Seventy Years”?
Nowhere. The seventy years ended in 539.