So, what happens if two gods have a child the normal way? Is the child a god or just a demigod (or maybe a demon lord/empyreal lord/ whatever)? Also what if a god and a mortal have a kid (As I imagine Cayden Cailean has done at least once)? I would think it would be a powerful outsider of the parent god's alignment (so a veranallia azata in Cailean's case.)
I don't think there's definitive rules about this, but they seem to range as far as power is concerned. For example, Dusk Ronin is a child of two full deities, but I don't think that she's anything other than a standard star archon (CR 19), Ragathiel (CR 26 demigod) is the son of Dispater (CR 27 demigod) and Feronia (CR ? demigoddess), so roughly the same power, but Shelyn (full deity) is the daughter of Thron (not any sort of deity at all (so presumably less than CR 25, possibly less than CR 20) and some other unknown creature. That's as many examples as I can think of off the top of my head.
It depends. Kurgess is suspected to be a child of Cayden Cailean and Desna, two full deities, but lived the life of a mortal before ascending to demigodhood. Ragathiel is an Empyreal Lord whose father is an Archdevil and mother is a demigoddess, and was divine from the beginning. Deskari is a Demon Lord because his parents were both Demon Lords. The children of mortals and Gods are called Godlings, and treated as gifted but otherwise ordinary people. Gods and demigods don't function the same way as mortals do, but exactly how or why isn't explained. This lack of specifics for exactly what happens when divine beings reproduce provides opportunities for plenty of interesting stories.
Planar Adventures lists three kinds of divine powers. Deity, Demigod and Quasi Deity. It also states that deities can have no worshipers and still be as powerful as one with trillions.
So the best answer would be, it's whatever works best for the story you're wanting to tell. Cayden Cailean and Desna raised the mortal Kurgess to demigod status because they were impressed with his deeds(or because he was their half-mortal son). But if you wanted Kurgess to grant more domains you could just say they raised him to a full deity or you could decide he was only given mythic abilities and hasn't quite made it to demigod.
Re_Kurgess:
Although he had been raised to demideity status by Desna & Cayden, he since may've become a full deity in that he grants five domains.
(Full deities grant five domains; demideities grant four; quasi deities usually grant less than but potentially up to four. Of course, this only applies to PF1: No indications of how this pattern will be affected/changed in PF2...yet.)
So, yeah...
Also has anyone created a homebrew god who is the child of two existing gods? I think that would be cool, particularly if the parent gods seem to be a somewhat unlikely couple (e.g Iomedae and Cayden Cailean, as despite one being L and the other C they both value bravery a lot.)
Shelyn and Zon-Kuthon are children of gods too, and they count as full gods, though they're much much older than Kurgess. Shub-Niggurath is established as an Outer God and has lots of offspring, sometimes with other Outer Gods, though none are as powerful as a deity. So I think it's safe to say that, at least at the outset, not offspring of two gods is automatically a god themselves.
I don't think there's definitive rules about this, but they seem to range as far as power is concerned. For example, Dusk Ronin is a child of two full deities, but I don't think that she's anything other than a standard star archon (CR 19), Ragathiel (CR 26 demigod) is the son of Dispater (CR 27 demigod) and Feronia (CR ? demigoddess), so roughly the same power, but Shelyn (full deity) is the daughter of Thron (not any sort of deity at all (so presumably less than CR 25, possibly less than CR 20) and some other unknown creature. That's as many examples as I can think of off the top of my head.Wasn't Thron originally a full deity before Zon-Kuthon beat him up and bound him as his herald ?
Thron was a spirit wolf who sired Shelyn and Dou-bral with an older love goddess.
The azlanti deity Sicva is the daughter of two other azlanti deities, Elion and Myr.
Also, Torag's wife Folgrit is a demigodess, as are all of their children and Torag's siblings.