Were any of Paul's letters already written by the time of the Acts 16 events?
No, none of Paul's letters had been written yet when the Acts 16 events occurred. The context places these events in the early 50s, before any Pauline epistle existed. His earliest letters, likely Thessalonians or Corinthians, were not written until about 58; Romans about 60; Ephesians, Philippians, Colossians about 64; and the Timothys and Titus about 66.
Les explains that the Acts 16 events occur while none of Paul's letters have been written yet—'not a one.' He places this in the early 50s and says Paul's earliest letter, probably the Thessalonian letters or maybe Corinthians, was not written until about 58. Romans follows about 60, Ephesians, Philippians, and Colossians about 64, and the Timothys and Titus about 66. So all these things are still taking place before Paul writes his epistles. Les also notes that once Paul has the revelation and is inspired to write his epistles, which become the Word of God, you no longer see as much of this early ministry phenomenon; for example, Paul's letters give no instruction for exorcism, and he was silent on casting out demons.