1. John saw a description of himself

2. John saw a description of him

Q1) If 'himself' and 'him' refer to 'John', does #1 sound like he emphasises that he saw HIS description?

3. Each student claimed that everyone but himself had been cheating.

Q2) Is there no other way to say except herself in the underline?
Q1 -- If both pronouns in #1 and #2 refer to John, then you cannot use #2.

Q2 -- All the students claimed that everyone but themselves had been cheating. Each student claimed that everyone else had been cheating.
Students: Are you brave enough to let our tutors analyse your pronunciation?
John read Mary's story about himself.

Q1) The sentence sounds odd to me. Is it OK?

Q2) If 'himself' changes to him, him may either refer to 'John' or someone else that Mary and John knows?
1-- Hard to say. Reflexive pronouns have two uses: to reflect the verb upon the subject, and as emphasis. This could be the latter. More context might show us whether 'himself' is a good choice.

2-- Of course... but then, so could 'himself'. Only greater context will show us the referents.