Yes, the centering argument of a projection can take the option Center. That uses the center coordinates of the country

where the projection is an argument of one of the cartographic type display properties.{"FullPolygon", "Polygon", "SchematicPolygon", "Shape"}

Projections, unfortunately, are not a possible argument of "CenterCoordinates" or other map coordinate related properties{"CenterCoordinates", "Coordinates", "FullCoordinates", "SchematicCoordinates"}

"Equirectangular" projection only moves the map along longitude direction. It is the same with "CylindricalEquidistant" and "Sinusoidal". For an example of a projection that centers, try "Albers":

Using this list of projections,
you can explore them to find one that does something reasonable.

Tooltips with data plots can be included in your graphics. Keep in mind, if you choose a projection that is something other than the default {"Equirectangular", {0,0}} then all of your decorations will also have to use the same projection. The alignment will be much simpler, sticking with one basic map rather than using unusual projections.

Also, keep in mind the map coordinates convention {lat, long} gets reversed when used as coordinates for graphics. For example the center of Greenland will be located at in order to get it over top of the Polygon for Greenland.