Magistrates

Where did Paul and Silas go before the magistrates' officers arrived the next day?
Since they later "went out of the prison" (Acts 16:40), they had gone back into the prison.

Who are "the magistrates" (Acts 16:35)?
στρατηγοι (strategoi), the original Greek word translated "magistrates," can mean either military officers in charge of military units or civilian officers who administer the law in a region. Since these magistrates "tore off their clothes" (Acts 16:22), they are civilian officers administering the law in Philippi.

Why were the magistrates "afraid" (Acts 16:38)?
They had broken Roman law. It was illegal to beat or bind a Roman citizen for any reason, yet they had done both and done them without a trial conviction.

Did Paul refuse to leave quietly out of pride?
That's highly unlikely. It's more likely that he did it to protect the small church that had just been planted. If he and Silas had left "secretly" (Acts 16:37) after having been falsely accused and beaten publicly, the Philippians could have concluded that the accusations and the punishment were justified, which would have invited more mob violence against the church that was gathering by the riverside. The original Greek verb translated "get us out" (Acts 16:37) is εξαγαγετωσαν (exagagetosan), which means to "lead out" or to "escort." Paul wanted the magistrates to personally escort them out of prison not only so that they - the magistrates - will be careful in dealing with Christians in the future, but also for the bystanders to spread the word that by personally escorting Paul and Silas out of prison, the magistrates had publicly admitted making an error against the Christians.

About what may the magistrates have "pleaded" (Acts 16:39) with Paul and Silas?
They probably apologized for having bound and beaten them, explained that they had not known that they - Paul and Silas - are Roman citizens, and begged them to not report them to their superiors.

How may Paul and Silas have responded?
Perhaps, "Ok, but we don't want to hear that you laid a hand on any other Christian in this city. Do we understand each other?"

After bringing Paul and Silas outside, why did the magistrates ask them to "depart from the city" (Acts 16:39)?
They may have been concerned that "the multitude" (Acts 16:22) may return to start another riot.

Did Paul and Silas heed the magistrates' request to leave the city?
No, they did what they wanted, and then left at the time of their own choice: they "entered the house of Lydia, and when they had seen the brethren, they encouraged them and departed" (Acts 16:40).

Why does Luke write that "they ... departed" (Acts 16:40)?
Luke stayed behind in Philippi, to rendezvous with Paul on his next visit.

When did that take place?
The Bible specifically mentions Paul spending eighteen months in Corinth ("And he continued there a year and six months, teaching the word of God among them" (Acts 18:11)), three years in Ephesus ("Therefore watch, and remember that for three years I did not cease to warn everyone night and day with tears" (Acts 20:31)), and three months in Greece ("... he came to Greece and stayed three months" (Acts 20:2)) before returning to Philippi. Given the months mentioned totaling almost five years, the other places Paul visited, and the travel in between, it most likely was six or more years later.

Did Timothy stay with Luke?
Since Acts chapter 17 records Paul leaving Silas and Timothy in Berea - "... but both Silas and Timothy remained there" (Acts 17:14) - Timothy departed Philippi with Paul and Silas.