Okay, so my theory is that if there is any history of a foreign trait in your sim's family tree, the sim has a 50/50 chance to get it. This chance does not further divide as the generation continues, as real genetics do.
Gen #1: Non-foreign---------------Chinese (two songs, uses chopsticks)
-Any child that this family produces will have a 50/50 chance to know both Chinese songs.
-Any child that this family produces will have a 50/50 chance to know how to use chopsticks,
Gen #2: Boy (two songs, no chopsticks); Girl (no songs, uses chopsticks)
-Since there is a history of Chinese, either of these children's offspring will also have a 50/50 chance to know both Chinese songs.
-Since there is a history of Chinese, either of these children's offspring will also have a 50/50 chance to know how to use chopsticks.
Ex. 1: Boy (two songs, no chopsticks)--------------French (two songs, French greet)
-Even though the Boy doesn't know how to use chopsticks, there is still a 50/50 chance his offspring will know how to use chopsticks, because it's in his history.
-The Boy knows two Chinese songs, so there is a 50/50 chance his offspring will know them, too.
-Any child that this family produces will have a 50/50 chance to know two French songs.
-Any child that this family produces will have a 50/50 chance to know the French greet.
Ex. 2: Girl (no songs, uses chopsticks)-------------Egypt (two songs)
-The Girl knows how to use chopsticks, so there is a 50/50 chance her offspring will know how, too.
-Even though the Girl doesn't know the songs, there is still a 50/50 chance her offspring will know how to use chopsticks, because it's in her history.
-Any child that this family produces will have a 50/50 chance to know two Egyptian songs.
Gen #3, Ex. 1: French Boy1 (two Chinese songs, uses chopsticks, no French songs, French greet)
French Boy2 (two Chinese songs, uses chopsticks, no French songs, no French greet)
French Boy3 (two Chinese songs, uses chopsticks, two French songs, no French greet)
French Boy4 (two Chinese songs, uses chopsticks, two French songs, French greet)
French Boy5 (two Chinese songs, no chopsticks, no French songs, French greet)
French Boy6 (two Chinese songs, no chopsticks, no French songs, no French greet)
French boy7 (two Chinese songs, no chopsticks, two French songs, no French greet)
French Boy8 (two Chinese songs, no chopsticks, two French songs, French greet)
French Boy9 (no Chinese songs, uses chopsticks, no French songs, French greet)
French Boy10 (no Chinese songs, uses chopsticks, no French songs, no French greet)
French Boy11 (no Chinese songs, uses chopsticks, two French songs, no French greet)
French Boy12 (no Chinese songs, uses chopsticks, two French songs, French greet)
French Boy13 (no Chinese songs, no chopsticks, no French songs, French greet)
French Boy14 (no Chinese songs, no chopsticks, no French songs, no French greet)
French boy15 (no Chinese songs, no chopsticks, two French songs, no French greet)
French Boy16 (no Chinese songs, no chopsticks, two French songs, French greet)
cousins
Gen #3, Ex. 2: Egyptian Girl1 (two Chinese songs, uses chopsticks, two Egyptian songs)
Egyptian Girl2 (two Chinese songs, uses chopsticks, no Egyptian songs)
Egyptian Girl3 (two Chinese songs, no chopsticks, two Egyptian songs)
Egyptian Girl4 (two Chinese songs, no chopsticks, no Egyptian songs)
Egyptian Girl5 (no Chinese songs, uses chopsticks, two Egyptian songs)
Egyptian Girl6 (no Chinese songs, uses chopsticks, no Egyptian songs)
Egyptian Girl7 (no Chinese songs, no chopsticks, two Egyptian songs)
Egyptian Girl8 (no Chinese songs, no chopsticks, no Egyptian songs)
So, I know that was really long, but I think there is a possibility to get all traits in one sim. And it wouldn't quite be as hard as originally imagined. You would only need one sim with a history of two foreigners and have that sim marry the remaining foreigner (ex. French + Chinese = Girl; Girl + Egyptian). It would take a lot of tries, but it might be neat to accomplish. Not for me, though. This diagram is as far as I want to go in trying that, lol.