The “render unto Caesar” episode does not negate the utility of the state. The temporal power is described as serving God in Romans 13, and therefore is a tool among many.
Encyclical is a letter sent by the Pope to the bishops of the Catholic Chruch and through them to the laity, that treats a given doctrinal matter. They are authoritative teaching, second in importance only to Apostolic Constitutions.
Obviously, a direct quote from Christ trumps a direct quote from any other author, in Christian theology.