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As you said, payment and shipping settings works (most of them) at website level. Noramlly you'll be able to change some label at store view level.
Store views are not supposed to be used to separate countries but languages. That's why you can edit those labels (to correct the text on each language).
Payment and shipping mehtods allows you to define some rules that depends on customer billing address or shipping address. That's how you can configure two shipping methods on the same website. Method 1 will work only with country A on shipping address and Method 2 with country B.
These rules depends on customer billing/shipping information. Once the customer enter the data Magento will refresh the available options.
You can share your costumers between websites.
I'm not sure about your business rules (why you want or need to separate countries per website).
Can you share your requirements about this configuration?
I'm still thinking that using different websites but sharing customers across all of them is the best idea here.