As long as we are giving civilizations special traits, lets go all the way.
France - Has to accept any offer made by Germany
China - all production 10% cheaper
India - loses a specialist to the US or England every turn
Germany - Must declare war on France at the beginning of each Age
Rome - civ spilts in two have way through the game
Aztec - loses a population to Spain during the first half of the game, loses a population to the US during the second half of the game
England - all tiles in London produce an extra gold (damn London real estate is expensive)
Russia - One unhappy person "disappears" each turn
US - Must run a budget deficit
Persia - can only nuke cities with Judism religion
Vikings - have to gift every specialist produced in the modern age to another civ (Noble Prize)
Inca - can import to civs with trade embargos and merc. civ
Carthage - Whenever you attack a country with elephants, one dies every other turn
Celtic - I got nothing, maybe something about Americans and tattoos?
Egypt - during the latter half of the game, your cities lose culture as it flows towards England and France
Greek - if either Greek or Ottoman found a city on an island the two countries are at a constant state of war
Ottoman - see Greek
Japanese - first nuke launched automatically lands on one of its cities
Malinese - you control the flow of gems
Mongol - if you become close to a domination victory, your leader dies and all your troops automatically return back to your capital
Spanish - every time you change state religions, every other religion is exsponged from your cities and your population drops
Arab - you must trade your oil to America, England, France, and Germany
Zulu - someone would just get upset about it