AnnOMaly:
Instead of renaming kings (without any support from history) and squashing them in in chronological order before Darius, it makes more sense to understand that the writer of Ezra began a giant parenthesis at 4:6 - as if he's saying, 'While we're on the subject of opposition to our temple rebuilding, I might as well mention what happened later in Xerxes' and Artaxerxes' reigns about our city rebuilding.' Then at 4:24 the writer resumes the narrative about the temple rebuilding.
If you relegate verses 6 to 23 as a parenthetical statement, you're left with (verses 4-5, 25):
4 Then the people of the land were continually discouraging the people of Judah and disheartening them from building. 5 They hired advisers against them to frustrate their plans all the days of King Cyrus of Persia until the reign of King Da·ri′us of Persia. ... 24 It was then that the work on the house of God, which was in Jerusalem, came to a halt; and it remained at a standstill until the second year of the reign of King Da·ri′us of Persia.
It was then that the work came to a halt? In the reign of Darius? That doesn't make sense. On the other hand, there's no good reason to distinguish the 'they' in verse 5 from the 'they' in the verses that immediately follow.
In addition, the attitude of Artaxerxes I (in Ezra chapter 7 & 8, and in Nehemiah) toward reconstruction in Jerusalem is entirely different to that of the Artaxerxes posited in Ezra chapter 4. There is nothing said of Artaxerxes I in Ezra chapters 7-8 or in Nehemiah to suggest that Artaxerxes I ever opposed reconstruction in Jerusalem; rather, he provides resources. In Ezra chapter 4, Rehum, the "governor of the region Beyond the River" is opposed to Jerusalem's reconstruction, but in chapter 7, the "governor of the region Beyond the River" is assisting with reconstruction. In Nehemiah, Sanballat and others objected after Artaxerxes I commissioned work in Jerusalem, but there is no reference to any associated request to Artaxerxes to halt repairs. Instead, work on the walls continues and is completed despite Sanballat's complaints to Nehemiah.