The gods themselves had an origin, thus "Theogony" was an accepted myth in Greek paganism.
Yes and no. Some were more cosmic than others, particularly Zeus.
Aside from uniquely northern European geographic features - realms of Fire and Ice - the Nordic gods were much like the Greco-Roman ones. Both had to contend with usurping Giants.
Well that's where your summary comes unstuck. The early Chinese religion was monotheistic and China had a few revivals of ethical monotheism from time to time, even prior to Western contact.
Most polytheisms have a Creator Spirit - the Egyptian pantheon for example was seen as aspects of the one underlying Spirit. So monotheism isn't odd, just usually de-emphasised in polytheisms.
Creator Sky-Gods appear through a multitude of human societies. I think they're a "strange attractor" in systems of explanation that rely on analogies to human agency. Hard-wired? Not so sure.