Why did the ancient Jews believe the messiah would be a great warrior who would establish the kingdom through military conquest?

The ancient Jews believed the messiah would be a great warrior who would establish the kingdom through military conquest because the people's expectation of a savior became bound up with the notion of a new nation, political kingship.

According to the Jews, was the Kingdom of God fully established?

According to the Jews, the Kingdom of God was not yet fully established to the Jews because so much evil obviously still existed in the world.

What are a few ways that the Jews had already seen God's kingship?

A few ways that the Jews had already seen God's kingship was in the "wonders of creation" and in the Torah.