My company only ships to a few countries but we would like to allow billing addresses from additional countries we do not ship to. How can I change the countries available in the dropdowns for shipping and billing addresses independently?
It is not possible by default to separate visible countries for delivery from a billing address. But it is possible for Magento to set specific shipping method restrictions. It means that if a person chooses a country where no shipping is allowed, he'll see an error message, that no shipping methods available at the moment.
That might be not something you expect your clients to experience, but that is the easiest way to restrict delivery, without creating customizations to address templates.
Also, you need to consider, that Magento has a very handy address book, and address from it is used for both shipping and billing sections, so even changing templates is not enough. In this case, I would stick to out of the box solution unless you have high bounce rate for shipping method step.