εὐημεροῦσαν goes adverbially with “ξυνοῦσαν”, ‘consorting in prosperity.’ The acc. is better than the nom., since it includes this condition in her prayer.
καὶ τέκνων (“τούτοις”) ὅσων κ.τ.λ.: ‘and with (those) of my children, from whom,’ etc.: “τέκνων” is partitive gen., the antecedent to “ὅσων” being understood. Cp. Plat. Rep. 387E “ὀρθῶς ἄρ᾽ ἂν...θρήνους... γυναιξὶ...ἀποδιδοῖμεν, καὶ...ὅσοι κακοὶ τῶν ἀνδρῶν”, i.e. (“τούτοις”) “ὅσοι”.
We might also take “τέκνων” as standing, by attraction, for “τέκνοις”. When the antecedent is thus attracted, it usually comes after the relative (O.C. 56); but sometimes precedes it, as in Tr. 151 f. “τότ᾽ ἄν τις εἰσίδοιτο”... | ...“κακοῖσιν” (for “κακά”) “ο<*>ς ἐγὼ βαρύνομαι”. In examples of the latter kind, however, the acc. (and not, as it would be here, the dat.) is elsewhere the case attracted. In favour of taking “τέκνων” as partitive, it may be added that this suits the distinction between two groups of “τέκνα”. By “ὅσων” she means Chrysothemis, Iphianassa, and the offspring o<*> her union with Aegisthus (157; 589).
ἐμοὶ … πρόσεστιν, attends upon me; cp. Tr. 453“ἐλευθέρῳ” | “ψευδεῖ καλεῖσθαι κὴρ πρόσεστιν οὐ καλή”.
