O righteous Father (Patēr dikaie). Nominative form with patēr used as vocative (cf. Joh 20:28), but vocative form dikaie. Then the righteousness of God is appealed to like God’s holiness in Joh 17:11.
The world (kai ho kosmos). The translations usually slur over the kai as untranslatable in English. Westcott suggests “while” as a sort of correlative. It is quite possible that here kai is almost concessive like “though” and de = yet: “though the world did not know thee, yet I knew thee, and these knew thee.” See Robertson, Grammar, p. 1182 for kai-de-kai and various other uses of kai in John’s Gospel.