See generally Kenneth Sacks in OCD(4) s.v. 'Diodorus(3)', and more fully in his monograph Diodorus Siculus and the first century (Princeton 1990). See also Oldfather's "General Introduction" at web address 1.
[1] "The Library" is only one version of the title of Diodorus' work; other sources refer to it as "The Library of History", "The Historical Library", "The Libraries." Of the 40 books only 15 survive intact, supplemented by copious fragments of the remaining 25. Diodorus' history is still the largest specimen of Greek historical writing to survive antiquity.
[2] Here the Latin Augustus is transliterated; the Greek would be Sebastos.
[3] For idiom cf. mu 46. (Sacks p.164 mistranslates e)pa/nw as 'afterward'.) The last dateable event mentioned in Diodorus' history seems to be 36 BCE. Since he mentions being in Egypt on business in the 10th Olympiad (60-57 BCE), it is unlikely that he survived Augustus (who died in 14 CE). The fact that the Suda records such a long lifespan has been taken to indicate that a later writer continued Diodorus' work into Augustus' reign.