1. Rhyf summarizes this pretty well.

2. The Vatican (Papal States) are a distinct political entity with all of the trappings and perquisites that acrue to any other state. Within the Vatican, the Pope is an absolute monarch and can rule by fiat. In practice, much of the day-to-day governance is run by the bureacrats and higher prelates of the curia. The Vatican allows Rome to provide much of its emergency service support, allows freedom of speech and the press, etc., but all of these practices are -- technically -- at the sufferance of the Holy Father. In practice, of course, attempts to reverse most of the policing and press policies at the Vatican are not so easily changed even if the Holy Father still technically rules by fiat.

3. The Holy Father is the head of the Church. He is tasked with service to the entirety of the faith and serves as the hierarchical leader of the clergy. His spiritual powers are, for the most part, no less and no more than those of any other consecrated bishop. The exception to this is, of course, that his interpretation of doctrine etc. is considered the final authority (at least until altered by some future Holy Father). The Pope's pronouncements are given great weight because of this in doctrinal matters. Only when the Holy Father makes a prounouncement that is specifically delineated as ex cathedra is he presumed to be articulating an absolute statement of belief applicable to the entirety of the Church. Such prounouncements are quite rare. All members of the clergy -- as indeed should all Catholics -- seek the guidance of the Holy Spirit in a continuting effort to grow closer to God.