for such little detail differences in meanings, you can always check books about Latin synonyms like Menge’s Lateinische synonimik. He mentions indeed that fundus indicates more the groud and also in regard to the fruits and vegetables you callect from it etc. If you want to indicate countryHOUSE villa is already sufficient. If you want you can also add the adjective villa rustica.
To indicate the totallity, I think you can use fundus, yes. I quickly went trhough a few examples of Cicero and indeed it seems fundus is used to indicate the totallity of a farm; but villa is also used to indicate the totallity. So I think you can choose, but normally you will always indicate more one of both: the fields or the building.