"Hello, Why does one say "the book of Job" and not "the book of Job's"?"
The word "of" can denote possession, and an apostrope can denote possession, but you must choose only one for one act of possession.
The examples you give are all a little different.
If you are talking about the Book of Job in the Bible, for example, it is common to describe it as "the book of Job". A Bible Study leader might say, for example, "turn to the Book of Job, or the Book of Acts, etc, etc." This usage is usual because some of the books of the Bible carry the name of the author while others do not, so a usage continues that can be applied for all of them.
If you bought a new book by Joe Citizen, you would say, "I bought Joe Citizen's new book."
"What about "a reader of Shakespeare" and "a reader of Shakespeare's" - are both correct?"
Only "a reader of Shakespeare". Here Shakespeare is a subject being studied. But you could say, "I am a reader of Shakespeare's work" and then you would show that the work you are reading "belongs to" or came from Shakespeare.
""A friend of John" or "a friend of John's"?"
Either "a friend of John" or "John's friend." I am a friend of John. I am John's friend. But you might hear people say "a friend of John's" because that sort of thing happens in speech, but it is not correct. :-)
""A follower of Mussolini" or "a follower of Mussolini's""
Either "a follower of Mussolini" or "a follower of Mussolini's ideas". Here the apostrophe in Mussolini indicates something possessed by Mussolini.