This was where the Roman soldiers in Jerusalem lived, and where the governor stayed when he was in Jerusalem. AT: "the courtyard of the soldiers' barracks" or "the courtyard of the governor's residence"
"the whole unit of soldiers"
Purple was a color worn by royalty. The soldiers did not believe that Jesus was king. They clothed him this way to mock him because others said that he was the King of the Jews.
"a crown made of thorny branches"
The greeting "Hail" with a raised hand was only used to greet the Roman emperor. The soldiers did not believe that Jesus was the king of the Jews. Rather they said this to mock him. (See: rc://en/ta/man/translate/figs-irony)