Today, I visited the Israel Museum in Jerusalem. It exposes a large model of that city during the reign of king Herod. The model shows a splendid theater of stone, just near the Roman quarters. The voice over at the model explicitely tells about the theater as a wooden construction. Although it doesn't give a full answer to the 'wood or stone' question, I stop my investigations on this.
The Second Temple of Jerusalem, during the reign of Herod the Great.
Herod, by the way, got his youngest son Jonathan, age seventeen, drowned by Herod's Gaul body guards in a swimming pool in Jericho. The boy did not seem to perform as a high priest during his first day in office. Herod the Pig! One of the many rather blunt episodes, while reading the Bellum Judaicum, or 'The Jewish War' of Flavius Josephus. But although this story has nothing to do with the topic of this blog, it catches the Zeitgeist of the era. For more stunning details: BJ I.431.