How to build select lists for search_api views
In this case I would create a Facet. You'll need to enable Search Facets (shipped with Search API) and Facet API.

Once you have that in place go to Configuration -> Search and metadata -> Search API and click edit on your Search index. In the dropdown there is a new choice called Facets now. 



Enable the facet for you field Project Countries. That field also needs to be indexed under your fields settings for your search index. Facet API will provide a block for you with the Facet that you need to place on your search page.

Facets work in a way that they only show options available in your search result. If "Sweden" is not present in the field Project Countries in any of the nodes you have indexed it will neither be available in the Facet. So facets work in a way that they narrow down search results and usually doesn't show options that will result in an empty search. That's the biggest difference I see between a facet and a normal filter in Views.

Facets are normally displayed as checkboxes or links (see display settings for your Facet). You mentioned in your question that you have it now in the View as a dropdown. You you want to keep it as a dropdown instead of checkboxes / links you could checkout Facetapi select