When having multiple country fields (billing country, shipping country, customer country) in a physical fact table, is there a need to duplicate the Geography Dimension to link to each of these country fields?

The Role Playing Dimension feature can be used to implement this scenario.  There should be one Geography dimension, and all the different country keys (Billing / Customer / Shipping) should be mapped to the Geography dimension.  While Querying from BI Tool (Tableau, Excel, etc.), it will be seen that the Geography dimension has been displayed three times (one each for Billing / Customer / Shipping).

In Design Center, provide a meaningful name to the joins when joining country keys to the Geography dimension. This name should be prefixed on the dimension name in your BI Tool.