If America is so bad for gods, why do they stay here? Surely just as many people or more believe in them from their original countries so why can't they go back? We know that when one of them dies he re-manifests in a different country, but I don't remember the book explaining why they don't just leave America.
I don't have the book here, but remember at the end :
Shadow meets Odin in Iceland
Which seems to implies that there's in fact one "different" God in every country visited by a worshiper of that God. American Gods are just a different form of Gods existing all over the globe. For example, the American Kali won't be very powerful, but her indian iteration (where there's a lot of people believing in Kali, even nowadays) will be totally different.
It's my interpretation.
IIRC, the gods became stranded in the new world because people stopped believing in them.
You see, the gods need (and stick with) people who fervently worship them and offer sacrifices to them.
So, the gods followed a minority of people to the new world and then when that minority died there was no one else to sustain them so some of them became stuck.
So, some gods get stuck because they need a human host (the more the better)
And the book also follows the hero's path or mythological cycle of birth, death and resurrection. So,
Some gods (like Shadow) are resurrected and some are reborn in other countries; but the impression I got was that there were multiple Odins because other cultures adopted them, reinterpreted them and made them their own. For example, the Romans conquered the Greeks but the Greek culture dominated the Romans which resulted in Greek and Roman versions of the same gods.