In terms of fairness, playing a set of 4 games with the same 4 countries is probably the best way to go, but I also think that making every game completely random could be a lot of fun, since between all the games we would get some pretty interesting country combinations. Since each game is only 2
and this is just a tournament to have fun, I would be more than happy to make each game completely random, but I would also understand if people would rather do the "more fair" approach.
If we are able to get 16 people, we have a lot of options for how to organize the rounds. We could have the top 2 from each set (8 in total) move on to a semi-final round, and the top 2 in each of those sets (4 total) move on to a final. Or if people don't want to play that many games, just have the top person in each group move on to a final.