I think for the Olympics, the medals belong to the individual athletes, and are recognized as belonging to the nation that existed at the time (so Yugoslavia, which won the gold medal in water polo in '88, its final Olympics as a nation, is still the "winner," instead of Croatia or another part of Yugoslavia).

That second part would apply to other sports, probably, but the trophy question's a tough one. I think the state with the most players is a better argument than the player with the victory goal (no "i" in "team," etc). But that's an easier call if one state dominates, tougher if three or four all have a number of players.

Another approach is to King Solomon it and just carve the trophy up. :-)