These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Anvilicious: Surprisingly averted, especially given the focus on the Catholic Church.
Genius Bonus: The story contains a lot of references to Church dogma which are clearer if the reader has a good working knowledge of Catholicism (and Judaism). Likewise, the use of dead languages.
I Am Not Shazam: There are some tantalizing hints that the Old Jew may be Leibowitz himself at first (note the Hebrew initials he scrawls on a rock - L.Tz.) but later in the story he denies this and explains he's only a distant relative.
The fate of Emily/Emma Leibowitz and the trapped people in the shelter Francis finds.
Abbot Zerchi trying to persuade a mother not to euthanize her child. The child is stated to be so badly burned by the atomic blast that her gender is not apparent until her mother refers to her as such. Also note that the mother is referred to as "the girl," (implying she is very young herself) and has also been diagnosed as so irradiated as to be untreatable.