What ideas about the Kingdom of God did Jesus clearly reject?

The ideas about the Kingdom of God that Jesus clearly rejected were that he was not trying to begin a new political reign over his people, and he was not trying to bring his opposition to any violence.

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.