So, I have some specific ideas on the topic of religions, here they are for consideration. Some or part of these have been suggested before by other players, but here I am putting it all together.
New religions
First of all, I was thinking of how to "classify" civ-specific religions into not-that-absurd "groups", so as not to have one religion per civ. I propose to add 5 religions to the game: Animism, three different forms of polytheism, and bringing back Judaism.
"Animism" (could also be "shamanism") would represent African traditional religions (for pre-Christian Ethiopia), Asian Animism-Shamanism (for pre-Confucian China's Shen and Wu (folk) religions, Korea's Muism, Japan's Shinto; Mongolia's Tengrism), and Polynesian animism.
I would've proposed to split the above in three groups (African, Asian, Polynesian), but seeing that the African and Polynesian groups are represented each by a single civ, one of which only appears for the player while the other (Ethiopia) is expected to become Christian for their UHV (while Mali and Congo spawn with missionaries), I didn't see much point.
Now, greek-roman polytheism and norse polytheism are all descendants of ancient "Indo-european polytheism", and they have a lot similarities, so I would propose to represent them as a single group. I would have the Persians and the Indians start with this religion as well.
As for the Middle East, a "Nilo-Semitic Polytheism" would cover the beliefs of the Egyptians, Babylonians and Phoenicians. There's of course no such thing in real life, the ancient middle east religions (Mesopotamian, Cannanite, Semitic, Assyrian) were very distinct from the ancient Egyptian religion, but otherwise Egyptian religion would be restricted to a single civ.
Judaism should spawn in any of these cities when monotheism is discovered by these civs (or the predefined date for Jerusalem), Christianity should spawn nearby, as should Zoroastrianism (unless its auto-founded by Persia). I realize Judaism is not even a one-civ religion, but it is a minority religion for many the civs, which is why I think it should be there.
Finally, the Mayans and Aztecs basically shared the same religion. Incan polytheism was related, but a little bit different. An "Amerindian Polytheism" would work for them all.
I realize that "Indo-european polytheism", "Nilo-Semitic Polytheism", and "Amerindian polytheism" are horrible names, so, any other name could work. I'm definitely not opposed to having Shinto, Egyptian Religion, Tengrism, African folk religion, Polynesian animism, and the Inca religions represented on their own, if grouping them with the other nearby religions seems weird, nor to having Shia Islam as a separate religion. I'm also not advocating to add them. Another idea is to have a second, civ-unique building to represent a couple of these (e.g.: "Egyptian temple", "tori gate"). I'll mention Sikhism and Jainism below.
Holy cities
As for the holy cities, I would propose to leave only 5 to 7 of them, specifically these: Jewish, Christian, Islamic, Hinduist, Buddhist, (maybe) Confucian, (maybe) Zoroastrian. Note that the Christian one would not be specifically Catholic, Orthodox or Protestant, but holy to all of them. Other religions (including Taoism) get no holy city (as they have no holy cities in real life… at least I don't think they do).
Temples
All religions have a unique "temple" building. All temples produce +1 happiness and +1 culture, but some have additional benefits (their hammer cost should also be different, animist shrines, for example, should be much cheaper than other temples). The following list is just illustrative:
Animist shrine: none.
Indo-european temple: +1 culture or +1 science?
Nilo-Semitic temple: +1 food or culture?.
Jewish synagogue: +1 commerce.
Zoroastrian fire temple: +1 science.
Christian church: +1 hammer.
converted to Catholic church: additional +1 commerce.
converted to Orthodox church: additional +1 science or hammer?.
converted to Protestant church: additional +1 science or hammer?.
Islamic mosque: +1 commerce, +1 science.
Buddhist stupa: +1 culture, +1 1 science.
Hindu mandir: +1 culture, +1 hammer.
Confucian pagoda: +1 science and (+1 commerce or +1 espionage).
Taoist pagoda: +1 health.
Amerindian altar: +1 hammer, +1 science.
Wonders & Cathedrals
The classical wonders that are currently linked to the Pantheon civic would be limited to the Nilo-Semitic and Indo-european polytheisms (some wonders, like the Parthenon only to the Indoeuropean polytheism, and others, like the Pyramids to the Nilo-Semitic religion), others can be to either (especially the Hellenistic ones, like The Great Library). The Pantheon civic itself could be changed to something else, for example a late-game (more or less forced) atheistic civic, which would go in nicely with communism and heavily secular countries.
Perhaps it would be interesting to see diplomatic-style wonders (like the Apostolic Palace) for Orthodoxy, Islam and/or Buddhism. These religions have institutions that have councils, determine doctrine, dictate rules, etc. Sikhism and Jainism could be represented with a wonder each (that have Hinduism as a prerequisite to be built).
I haven't given much thought to Cathedrals (and their equivalents), but I would propose, for the moment, that they can still be built by any civ that has 4 temples of the religion (for all religions).
Monasteries and Religion Spread
Monasteries would be restricted to religions that actually have monastic culture and/or religions that actively evangelized: all Christian religions, Islam, Buddhism, Hinduism, Taoism, Confucianism. The other religions (Animism, Indo-european, Nilo-Semitic, Amerindian, Zoroastrianism, Judaism) automatically spread to cities when they are founded if they are the state religion.
For example: China should start as "Animist", so their first cities have what we call "Chinese folk religion" automatically. When Confucianism is founded, China can choose to take it as state religion and then the automatic spread of Animism to newly founded cities stops; at that point Animism will not be deleted from cities where it is present and it will continue to spread through the current mechanism of ("random") religion spread. Same thing goes for Persia and India starting with Indo-european polytheism in their cities until they found Zoroastrianism (which I think shouldn't be upon-spawn as it is now) and Hinduism.
Now, since many of the smaller religions aren't practiced anymore (or minimally), I would propose for a mechanism that makes religions disappear just as they can spread. This would apply to all religions (including the "major" ones that are already in game), but with different probabilities depending on the religion, date and geography. As proposed elsewhere, inquisitors would be able to wipe out religions (player chooses which) from cities in your empire and outside of it (provided civics like Theocracy aren't present). However, no state-religion can disappear from your cities, and religions have less chance of disappearing under liberal civics (free religion, etc), but more chances of disappearing under "atheism" as civic.
State Religion & Civics
Now, all religions should provide different benefits when they are the state religion. Things such as extra specialists for Indoeuropean polytheism, extra research or food for Amerindian polytheism, extra commerce routes for Judaism, extra research rate for Confucianism, etc., could be some ideas.
Under certain religious civics, some religions should lose access to some resources (i.e., under more "orthodox" civics as opposed to more "liberal" civics). Islam should have no access to pigs, wine and tobacco; Judaism to pigs and seafood; Buddhism to wine and tobacco; both Hinduism and Buddhism to cows, sheep, pigs (and fish?).
Founding religions
As animism, indo-european polytheism, nilo-semitic polytheism, and amerindian polytheism should automatically appear with cities as they are founded (including the capital, before techs can be researched) I would propose to link them to very early technologies (perhaps even to agriculture or animal hunting) or I would propose that they have no tech prerequisite. Given that, I would change the name of the current "Polytheism" tech to something else and eliminate the "Monument building" tech altogether. Hinduism would appear then not with the tech "Polytheism" as it does now, but perhaps with "Priesthood". A non-divided Christianity should spawn originally, with an event triggering the Catholic-Orthodox schism and a second event triggering the Catholic-Protestant schism.
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I hope I didn't miss anything and that all was clear. Obviously all numbers (e.g.: x temple produces y hammers) have to be balanced, as well as building costs and the religion-specific points (such as state religion benefits). So... thoughts on all of this?