For all we know there very well could still be a Pope in Rome. We have no idea.
As for how communication is possible. Tenpenny made the journey to America. Its possible that trans-atlantic travel is occasionally possible. Perhaps a ship carrying an apostolic nuncio occasionally makes the trip to America to inform the East-Coast based church of Vatican affairs.
Not necessarily. In several instances throughout history the Pope has not been located in Rome. The most important aspect of the Pope is apostolic succession from St. Peter, not his location.
Besides, the tenets of the Catholic faith and following the catechism would be the premient aspects of them being considered "catholic." Which as far as we know, there's nothing about Marcella, Father Clifford, Diego, or any of the Rivet City residents that breaks with catholic tradition. Both Father Clifford and Diego are celibate for instance. Marcella recites an act of contrition. They speak of "breaking bread" (likely communion) and Father Clifford speaks of St. Monica, who is probably a canonized post-war saint.
To my knowledge they do nothing that IRL Catholics don't do.