I read somewhere else that Neri was a capo in part III. Is that true?

I do wish the Lucy Mancini background was shorter!!!

You and me both.
Puzo was one of those writers (like Ian Fleming) who couldn't resist passing along in his novels any and all specialized knowledge he'd picked up in his experience. Fleming used this technique to great effect in the Bond novels, but Puzo was ham-handed. It's pretty clear to me that some female of Puzo's acquaintance had had the condition and the operation; that Puzo interested himself in it--and couldn't resist passing along the whole sorry description in "The Godfather." So, it seems, he created those endless Jules-and-Lucy chapters for the sole purpose of permitting himself to describe her operation.

Johnny Fontaine serves a very useful purpose in the beginning to illustrate what it meant for him to have Vito as his godfather; and to set the stage for the excellent Woltz and horse-decapitation scene, which illustrated Vito's ruthlessness. Thereafter, Fontaine (and his pal Nino) clutter up the novel for one purpose: to allow Puzo to show off all the stuff he learned in Hollywood before the novel was published. Thank goodness FFC left most of that out in the film. The main reason, IMO, that the TV movie of "The Last Don" was so superior to the novel was that the movie left out most of the Hollywood crap that wrecked the novel.