If you could separate them into separate screens than having Religions, Cults, and Guilds would work nicely (CotD would obviously become a cult instead of a religion). Having Guilds and Cults intermixed on the same screen would just be wrong. I would prefer a separate cult screen, but they could probably just be arranged in two separate rows at the top and bottom of the same screen.
Cults might be like unadoptible religions that probably cause more harm than good except in their holy cities (They should also have the same negative penalties as non state religions from civics, and be wiped out by inquisitions). It might be nice if they spread faster than real religions, but sometimes died out on their own. Some cults could also have some synergy with some religions and huge problems with others, possibly based on alignment. I'm reminded of how the Civilopedia says that the Lugus and Sirona religions are tolerated by the Order, but their priests are required to give assistance as healers, exc. Perhaps their current priests could get fewer spell spheres, but also get the spheres of the more synergistic cults. For example, Confessors might start only with Law (and probably fire, unless both a Bhall and Brigit cult are added) magic but have access to Spirit, Life, and Sun sphere if built in cities with the Sirona, Sucellus, and Lugus cults. Esus and Mammon cults might let evil/neutral disciples have access to shadow and mind magic. Would it be too much to give each civ a cult for its representative elements, and make that civ immune to its negative effects? It might be best to let anyone found the cults, but make it much easier for them. Since several wonders are related to certain gods, it might make sense for these to require the cult (or to found the cult if that is a better mechanic) It might also be appropriate to make cults have unique units, or to make current units like Monks or Shadows dependent on certain cults. Since the cults are not government sanctioned, it might be nice if their units were not buildible (except perhaps by the founder or the civ that represents that cults element), but would sometimes randomly join your cause.
Guilds would be much more like the Cooperations in BtS. They would probably have purely economic, non-religious effects. Guilds would be improved by the Guild civic. I'm undecided as to whether tieing the current Mage Guild and Theves Guild buildings to actual Guilds is a good idea. If so, perhaps the Mage guild should be founded at the start of the game by the Amurites when they are present, and whoever gets Knowledge of the Ether first otherwise. If they do decide that each civ should start with a cult, then perhaps some less magically inclined civs (like Khazad and maybe Hippus) could start with a guild instead.
but my understanding is that corporations allow you to trade one or more resources for other resources and/or happiness, etc. So, cows, sheep, corn, rice, horses, etc. 
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9 (decreases Pop cap)
