With regards to papal infallibility, it is a negative power that has a strictly defined use:
-The Pope must teach in his public and official capacity as pastor and doctor of all Christians, not merely in his private capacity as a theologian, preacher or allocutionist, nor in his capacity as a temporal prince or as a mere ordinary of the Diocese of Rome. It must be clear that he speaks as spiritual head of the Church universal.
-Then it is only when, in this capacity, he teaches some doctrine of faith or morals that he is infallible.
-Further it must be sufficiently evident that he intends to teach with all the fullness and finality of his supreme Apostolic authority, in other words that he wishes to determine some point of doctrine in an absolutely final and irrevocable way, or to define it in the technical sense. These are well-recognized formulas by means of which the defining intention may be manifested.
-Finally for an ex cathedra decision it must be clear that the Pope intends to bind the whole Church. To demand internal assent from all the faithful to his teaching under pain of incurring spiritual shipwreck (naufragium fidei) according to the expression used by Pius IX in defining the Immaculate Conception of the Blessed Virgin. Theoretically, this intention might be made sufficiently clear in a papal decision which is addressed only to a particular Church; but in present day conditions, when it is so easy to communicate with the most distant parts of the earth and to secure a literally universal promulgation of papal acts, the presumption is that unless the pope formally addresses the whole Church in the recognized official way, he does not intend his doctrinal teaching to be held by all the faithful as ex cathedra and infallible.
The number of ex cathedra decisions defined dogmatically throughout the history of the Church is a matter of some debate, but is relatively small given the near 2,000 year history of the Church. A Catholic theologian named Klaus Schatz, in a study released in 1985, listed seven.
More details here: http://www.newadvent.org/cathen/07790a.htm
With regards to the Vicar of Christ title, it is symbolic of the Pope’s primacy in jurisdiction over the universal Church. A “vicar” in general is someone who is acting as the agent of a superior; it has been used for ecclesiastical positions other than that of the Pope, but the particular titles associated with the pontiff over the centuries include “Vicar of St. Peter”, “Vicar of Christ”, and “Vicar of the Apostolic See”.
And in the long history of the Church, there have been some truly rotten Popes: Alexander VI. John XII. Stephen VI. Leo X. Benedict IX. Urban VI. Clement VII. I’m sure there have been a few others.
Yet in the course of human history, rife with weak and sinful people, entire nations, peoples, ideas, and languages have risen and fallen, while the Church remains standing. Given how rotten humans can be, that’s a miracle in and of itself.
The Church will survive Pope Francis.