Is it permissible to read books about Avodah Zarah?
Is it permissible to read Greek mythology for informational purposes, seeing as it's not particularly tempting to anyone nowadays?
Is it permissible to read novels about Avodah Zarah, like Percy Jackson and the Olympians, which is about Greek gods?
Is it permissible to read novels of kefirah, such as Phillip Pullman's His Dark Materials series, where God is killed?
The only question that I'm going to answer directly is number 2, since I heard directly from my Rebbi that it is 100% permissible (unfortunately, I can't quote it in his name since I didn't get his permission to use his name on this site, but I'll say that he's a well respected Musmach from Yeshivas Chafetz Chaim). He said that given the limited number of kosher 'outlets' available nowadays, we don't care as much about Avodah Zarah [especially given the lack desire nowadays to serve idols], so as long as the book is appropriate in other matters, it can be read.
I assume that this answer would also apply to the book you mentioned in question 3, but I'm writing that separately, as the specific example of Percy Jackson was asked to my Rebbi, so I can confirm that question with certainty. It might even apply to your first question also, but that case might be different. I wouldn't be too surprised if there's a difference between informational purposes and entertainment purposes.
Reb Moshe has a Teshuvah on a similar topic (Y.D. Chelek 2, Siman 53. It's a fascinating Teshuva, and relatively short too (only about a column and a half), if anyone's looking for a short, fun piece to read). A history teacher asked if he was allowed to teach about Greek history in public school, as it would necessarily include speaking about Avodah Zarah. Reb Moshe responded it was permitted, since everyone nowadays knows it to be false. He also mentioned that it might even be a good lesson for the students, to show them that it's possible for a majority of the world to believe in something false.
It's interesting to note also that Reb Moshe wrote that if a textbook was written with the intent of glorifying the idol, it is forbidden to read it even nowadays, while if it was written with the intent of mocking the idol (i.e. describing the 'crazy things' that idolaters do), then it was permitted to read it even in the days when idolatry was prevalent. Yet it's curious that Reb Moshe doesn't directly address the case where a work was written for 'neutral' reasons, i.e. educational purposes.
It's also important to note that Reb Moshe's Teshva was addressed to a teacher, who needed to teach that information as part of his livelihood (granted, Avodah Zarah is one of the big three sins that one would need to give their life for and not transgress, all the more so one would need to give up their job, but if Reb Moshe's question was dealing with a different transgression [and from the context of the Teshuva, it seemed like the issur in question was only a rabbinic decree to prevent 'Al Tifnu El Haelilim', which unfortunately I don't know enough about to explain it's relationship to 'Avodah Zarah], then the permission might only apply in the teacher's case).
Although there is a prohibition to read books about idolatry or even say their names (Rambam Avodah Zara 2:3), R. Moshe Feinstein (Y.D. II 53) has stated that, like in other halakhos of Avodah Zara, we need not be concerned if that form of idol worship has been annulled (which in this context means that nobody worships it anymore). However, R. Yair Hoffman believes that even Rav Moshe would prohibit reading about idols and idolatrous practices unless it is portrayed in a negative light.
While there's no prohibition in reading about fictional 'gods' per se, reading novels is not a simple matter. The Yerushalmi (Sanhedrin 10:1) states that reading'books of Homer' is like reading letters and permissible, but most poskim imply otherwise. The Shulchan Aruch (O.C. 307:16) seems to prohibit reading all novels as does Tosfos and Rosh to Shabbos 116b, and the Rambam (Commentary to Mishnah Sanhedrin 10:1) is particularly emphatic that one who reads books of stories, "kings and their behaviors", wastes time and doesn't deserve a share in the World to Come. These days, it seems that many Rabbis take it for granted that we need some form of entertainment, and it's probably better than doing nothing, as בטלה מביאה לידי זימה (idleness brings about lewdness - Kesubos 59b), but this is a sensitive matter and should probably be brought to your LOR (local Orthodox Rabbi). More importantly, contemporary novels (even ones written for teenagers) may contain what many b'nei Torah consider to be 'inappropriate material'. (I wouldn't know about this series as I've never read them, sorry).
I'm not entirely sure if I understood from Wikipedia what's considered a 'novel of kefirah', but when it comes to books that one might suspect would lead one away from God there's the explicit prohibition of 'אל תפנו אל מדעתכם' - do not remove God from your minds (Shabbos 149, see Igros Moshe quoted above). While there may be dispensations in order to know how to respond to a heretic (Rambam, Rashbatz and R. Yakov Emden to Avos 2:2, Meiri to Sanhedrin 11:1), I would still ask your LOR here to: the answer may be different depending on the individual book as well as the personality (and possibly level of religious commitment) of the person asking the question.
