The Foreign Office spent about two weeks before this note was published making sure that the Zinoviev letter was really genuine and did come from Moscow.
The British and Soviet governments reached many minor agreements before 1923. One of them was the formula on propaganda in May 1923. This basically meant that the Soviets agreed not to support organisations which stirred up trouble in Britain.
The Third International basically means the same thing as the Cominform - the way Moscow tried to spread Communism around the world.