(Question 1) Did the Eunuchs have intimate relations or criminal conversation with King Herod's son Alexander in Antiquities, Book XVI, Chp. 8?
The passage says in Greek:
Whiston's 18th century translation says: "there was one told the king that these eunuchs were corrupted by Alexander the king's son with great sums of money. And when they were asked whether Alexander had had criminal conversation with them, they confessed it, but said they knew of no further mischief of his against his father".καί τις ἀγγέλλει τῷ βασιλεῖ διαφθαρῆναι τούτους ὑπὸ Ἀλεξάνδρου τοῦ παιδὸς ἐπὶ πολλοῖς χρήμασιν. ἀνακρίναντι δὲ περὶ μὲν τῆς γεγενημένης πρὸς αὐτὸν κοινωνίας καὶ μίξεως ὡμολόγουν, ἄλλο δὲ οὐδὲν δυσχερὲς εἰς τὸν πατέρα συνειδέναι.  βασανιζόμενοι δὲ μᾶλλον κἀν ταῖς ἀνάγκαις ὄντες ἐπιτεινόντων ἀεὶ τῶν ὑπηρετῶν καὶ χαριζομένων τῷ Ἀντιπάτρῳ τὸ τοιοῦτον, ἔλεγον ὡς εἴη δυσμένεια πρὸς τὸν πατέρα καὶ μῖσος ἔμφυτον Ἀλεξάνδρῳ.
In Loeb's 1920 translation, the Eunuchs confess to "intimate relations" with Alexander: "When Herod asked whether they had had intimate relations with Alexander, they confessed to this but said that they were not aware of any other offence on his part against his father."
In her book Jewish Slavery in Antiquity, Catherine Hezser takes this quote from Loeb's edition to refer to sexual relations.
And "intimate relations" apparently can mean a sexual relationship, because in Loeb's edition, because in the story of Ida getting Munda to have a relationship with Paulina (Ant. VIII.66-70), Loeb's translation uses this term:
However, perhaps there could be be ambiguity in the Greek terms. By comparison, the word "intercourse" in English can mean either sex or conversation.She went to him, used argument to rouse him, and bu plausibly undertaking to find a way, held out hope that he might succeed in enjoying intimate relations with Paulina.
(Question 2) Did Josephus say that God appeared to the Pharisees or that He inspired them?
Whiston's translation says that Pheroras' wife paid a fine that was laid on the pharisees, and in return they gave her a favorable prophecy with divine inspiration:
Loeb's translation however says: "In return for her friendliness they foretold - for they were believed to have foreknowledge of things through God's appearances to them - that by God's decree Herod's throne would be taken from him...."In order to requite which kindness of hers, since they were believed to have the foreknowledge of things to come by Divine inspiration, they foretold how God had decreed that Herod's government should cease, and his posterity should be deprived of it; but that the kingdom should come to her and Pheroras, and to their children.
I think that this is the sentence in the Greek: "οἱ δὲ ἀμειβόμενοι τὴν εὔνοιαν αὐτῆς, πρόγνωσιν δὲ ἐπεπίστευντο ἐπιφοιτήσει τοῦ θεοῦ, προύλεγον, ὡς Ἡρώδῃ μὲν καταπαύσεως ἀρχῆς ὑπὸ θεοῦ ἐψηφισμένης αὐτῷ τε καὶ γένει τῷ ἀπ᾽ αὐτοῦ, τῆς δὲ βασιλείας εἴς τε ἐκείνην περιηξούσης καὶ Φερώραν παῖδάς τε οἳ εἶεν αὐτοῖς. "
If God was still making appearances to the pharisees in Herod's time, it makes me wonder what those appearances or theophanies were like.
(Question 3) Can you make sense of the ending of the Greek sentence below?
Josephus writes about Antipater's detention: "And after putting him in chains, Herod sent out a letter about him to Caesar in Rome and also sent some men to inform him by word of mouth about the villainy of Antipater." According to the translator Ralph Marcus, the Greek here has an additional, unintelligble part, underlined and beginning with "kai Koponiou" below:
δήσας δὲ αὐτὸν εἰς Ῥώμην ὡς Καίσαρα ἐκπέμπει γράμματα περὶ αὐτοῦ καὶ τοὺς ἀπὸ γλώσσης διδάξοντας τὸν Καίσαρα τὴν κακίαν τοῦ Ἀντιπάτρου καὶ Κωπωνίου γνώμη τὴν Καίσαρος.
(Question 4) What does the Greek name "Damneion" mean? I guess that the name means "The Condemned". What 1st century Jew would be named "The Condemned", especially with a name in Greek, rather than in Aramaic or Hebrew?
Here is how Josephus tells the story of James' killing, leading to the appointment of Jesus Ben Damneus as High Priest:
The figure of the high priest Jesus son of Damnaeus, who replaced Ananus, is a mystery for me....this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, (23) who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity [to exercise his authority]. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrim of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king [Agrippa], desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrim without his consent. (24) Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest.
I don't know if this was a fictional character, but he shows up a few places later in real circumstances. For example, in the next chapter, Josephus writes: "the high priest, Ananias... increased in glory every day, and this to a great degree, and had obtained the favor and esteem of the citizens in a signal manner; for he was a great hoarder up of money: he therefore cultivated the friendship of Albinus, and of the high priest [Jesus], by making them presents".
Josephus talks about Jesus ben Damnaeus again here:
And now Jesus, the son of Gamaliel, became the successor of Jesus, the son of Damneus, in the high priesthood, which the king [Agrippa] had taken from the other; on which account a sedition arose between the high priests, with regard to one another; for they got together bodies of the boldest sort of the people, and frequently came, from reproaches, to throwing of stones at each other. But Ananias was too hard for the rest, by his riches, which enabled him to gain those that were most ready to receive.