(Question 1) What did Seneca probably think about Christianity?
The surviving correspondence between Seneca and Paul is probably counterfeit, so he did not make open public statements about it. We don't know if he accepted or rejected Christianity, but he probably didn't accept its theology because the Stoics respected the Roman gods. Granted, he could have been a secret Christian who kept silent about it due to fear of persecution. But the number of secret Christians, and Christians in general, in his time appears to be only a tiny fraction of the Roman populace.
Nonetheless, Seneca should have been aware of Christianity, since Paul in Philippians sends greetings to those of Nero's household in general, and particularly Epaphroditus, Nero's secretary.
Plus, Seneca probably sympathized with it philosophically and socially. His brother Gallo effectively in effect defended Paul in Corinth in Acts 18. Here is the background:
Here is the passage in Acts 18:Some of Paul’s activities had led to riots in the provinces and his name repeatedly came to the attention of the Roman authorities. One such incident, four years earlier, involved Seneca’s brother Gallio, then serving as a magistrate in Greece. The Jews of Corinth had charged Paul with spreading false doctrines and brought him before the magistrate’s court but Gallio refused to hear the case.
https://www.metrum.org/gosen/fromtraggo ... m#_ftnref8
This account sounds realistic. In Josephus I read many accounts where Syrians, Greeks, and Jews go to the Romans to settle disputes with each other, and the Romans' decisions are unpredictable in that they don't automatically side with one group over the other. There were also numerous Hellenistic Jews in that era in the Levant who had close relations with the Greeks. That being the case, it's easily foreseeable that the Romans wouldn't punish Paul for teaching in contradiction to the Torah.12. And when Gallio was the deputy of Achaia, the Jews made insurrection with one accord against Paul, and brought him to the judgment seat,
13. Saying, This fellow persuadeth men to worship God contrary to the law.
14. And when Paul was now about to open his mouth, Gallio said unto the Jews, If it were a matter of wrong or wicked lewdness, O ye Jews, reason would that I should bear with you:
15. But if it be a question of words and names, and of your law, look ye to it; for I will be no judge of such matters.
16. And he drave them from the judgment seat.
17. Then all the Greeks took Sosthenes, the chief ruler of the synagogue, and beat him before the judgment seat. And Gallio cared for none of those things.
The introduction to a modern edition of Seneca's Letters from a Stoic, Epistulae Morales ad Lucilium (Selected and Translated with an Introduction by Robin Campbell, Penguin Books, Baltimore ● Maryland, 1969) notes the parallels between Seneca's philosophical and moral teachings and those found in the New Testament. For example, Whereas Paul writes: "The love of money is the root of all evil (1 Tim. vi. 10)", Seneca says in ("On Tranquility of Soul", 8 ), "Riches [are] the greatest source of human trouble." Whereas Paul writes: "Be kindly affectioned one to another with brotherly love." (Rom. xii. 10), Seneca writes "Man is born for mutual assistance." (On Anger, i.5) Matthew v.45 says: "He maketh the sun to rise on the evil and on the good", whereas Seneca writes (On Benefits, i, 1): "How many are unworthy of the light! and yet the day dawns."
(Question 2) What do you make of Eusebius giving "Seneca" as the name of the tenth bishop of Jerusalem? Was it a common name in the Roman empire? Or maybe this was a Christian who, on converting to Christianity, took on the name of Seneca in honor of the philosopher in a way like Saul received the name of Paul or like Simon took on the name Peter?
The major Fourth Century bishop Eusebius listed "Seneca" as the tenth bishop of Jerusalem's church in his book "Church History". This stood out to me because Jerusalem's first ten bishops served in the 1st to early 2nd century and were known to be of Jewish background (or more precisely, Torah-observant). Eusebius writes:
(Question 3) If Seneca's De superstitione is lost, what basis do writers like Kenneth Humphreys or Paul Davis have for their assertions that Seneca, and this work in particular, didn't mention Jesus?3. But since the bishops of the circumcision ceased at this time, it is proper to give here a list of their names from the beginning. The first, then, was James, the so-called brother of the Lord; the second, Symeon; the third, Justus; the fourth, Zacchæus; the fifth, Tobias; the sixth, Benjamin; the seventh, John; the eighth, Matthias; the ninth, Philip; the tenth, Seneca; the eleventh, Justus; the twelfth, Levi; the thirteenth, Ephres; the fourteenth, Joseph; and finally, the fifteenth, Judas.
Kenneth Humphreys asserts: "A lost work De superstitione [by Seneca] ridiculed popular conceptions of the gods... Seneca... was contemporaneous with the 'Jesus' of legend. Yet though Seneca wrote extensively on many subjects and people, nothing relating to 'Jesus' ever caught his attention." (http://www.jesusneverexisted.com/seneca.html)
In his essay "Christian foundation crumbles under scrutiny", Paul Davis asserts: "Seneca the Younger, in his book On Superstition (where he criticizes every known cult and religion) makes no mention whatsoever of Jesus or Christianity." (https://ffrf.org/publications/freethoug ... paul-davis)