No, popes are only infallible when speaking ex cathedra on matters of faith and morals. In all other matters they are no different from any other human.
Yes, the Catechism of the Council of Trent, also known as the Roman Catechism, is considered authoritative and reliable teaching by the Catholic Church, but it is not considered infallible in the same way that papal or conciliar pronouncements are. It does provide a comprehensive and orthodox presentation of Catholic doctrine.
yes, unfortunately there were some popes who were morally corrupt. but not in matters of faith and doctrine.Yes, definitely. As an example, do some reading on "Inquisition", which was carried out by the Catholic Church, under several popes.
Yes, they had many differences on matters of authority and doctrine. Some Popes were even excommunicated by Church Synods (or Councils) and other rival Popes were elected in their place, called Ante-Popes.
Gelasius
The first 16 popes were martyred for their faith.
If they believe that the pope is teaching them that Jesus Christ is the way to salvation then yesIf you are asking if the pope is an agent of salvation then the answer is no.
The problems of the church and the fights between the Popes and Kings of assigning bishops. Also the crusades.
He believed Christians should be saved through faith and popes lacked this authority.
Popes do not elect popes. Popes are elected by the cardinal electors.
No. It is literally impossible for a woman to be ordained, these has been the constant teaching of the Church and has been reiterated by the most recent popes. Thus it is an infallible teaching and no future pope could reverse it.
The majority of popes started as priests. However, it is not a requirement. Any male Catholic can be chosen. Pope Benedict XVI had virtually no experience as a diocesan priest as he spent most of his time in academics - teaching in universities - and was promoted directly from there to archbishop.