Yes, "mongoloid" is a highly offensive and outdated word. However, it doesn't correlate with the modern understanding of Mongol, Mongolian or Mongolia (unless you actively choose to be backwardly racist). I think this is where your misunderstanding lies.

I know which quote from Prison Break you mean, and the speaker (Bellick) is being as offensive and disgusting as possible. Even in that quote, there's no indication that the word is acceptable in any way.

The original name, still commonly used when I was a kid in the 1950s, was indeed "mongoloid idiocy." It wasn't used because Mongols were thought to be of low intelligence, but because people thought that people with Down's syndrome physically looked like Mongols. To quote a book from 1922 that I found, "the face is characteristic: slanting red eyes, coarse hair, and blunted occiput," and another book says "In mongoloid idiocy the child has a peculiar physiognomy, which has led to the name mongoloid." 

You say "I don't think it's appropriate to name such a disease after a typical nationality." You are right. It isn't. In the U.S. the phrase is now considered to be offensive and it isn't used any more, and hasn't been for decades. The term was undoubtedly familiarized in the United States by the TV situation comedy, "Life Goes On." It aired during the late 1980s and included a character with Down's syndrome (played by an actor with Down's syndrome).

I do not understand why anyone would downvote Katarina´s response, unless they do not understand why it is relevant in this case, (I personally upvoted it back up to zero.)  

The issue is that somehow the term mongoloid was used for children with Down Syndrome and that is the context that many people learned it.  It was the way I learned it as a child.  I did not even know until going to look it up now that ¨mongoloid¨ was a term for anything else.  

Here are the two definitions from the dictionary that comes up when I searched Google: 1. ¨of or relating to the broad division of humankind including the indigenous peoples of eastern Asia, Southeast Asia, and the Arctic region of North America.¨ 2. ¨having Down syndrome.¨        I cut enough for the character limits, which include notes that the first is dated and offensive and the second is offensive.    

I agree it was not appropriate or at all culturally sensitive to link Down Syndrome to a group of persons from Asia, but in the time this word was coined, people did not have the same standards as we have today.  The world did not have so much global interaction back then.  I agree it is a word that is better not to use--- if the dictionary I used is correct that both definitions are offensive. 

Damon,

Words like retard and idiot are often used in very informal conversations or within close friends.  Otherwise, they are highly disrespectful words.  "Mentally challenged" is probably a better term to use here.

As an English learner, I find the cultural context of a word is often time the most difficult part for us.  In an English-Chinese dictionary, these words/terms probably all mean the same thing.  While watching these TV series to learn a language, we really should be sensitive to not just what words are being used, but how, when and where they are used.

Eddie