Is the Chinese character "蒙" being associated with Mongolia and ignorance coincidence? In the translingual section, Wiktionary mentions that the Chinese character 蒙 has a variety of meanings:


to cover
ignorant
suffer
fourth hexagram of the I Ching
Mongolia - radical composition becomes 'grass(land) family'
a surname


Are meanings 2 (or any other meanings) and 5 considered related?

I suspect that meaning 5 having a radical composition of grassland family would suggest not, but I don't know.

I also once read in a discussion on so-called "Mongolian beef" someone claim (without any references) that in Chinese, "Mongolian" can mean stupid, or just used as a derogatory term:


  First no offense to anyone, esp. people from Mongolia.
  
  Mongolian in Chinese can either mean  a. people from Mongolia b. not
  smart, retarded
  
  If someone is called a "mongolian doctor" it means that his medical
  practice is sub-par at best.
  
  I believe in the case of mongolian beef, it simply means untasty beef
  - in a twisted sense of chinese humor. Although as a Chinese, I find mongolian beef actually not tasty.


Is this true?

Also, would I be correct in concluding any association between Mongolia and stupidity isn't because of Down Syndrome? The Chinese edition of Wikipedia has an article on Down Syndrome, and it mentions 「蒙古症」and「蒙古痴呆症」, but says that "国际人" is more common (though that doesn't sound particularly politically correct to me).
亚 means inferior, hence Asia is an inferior continent? No, it has to do with approximating sounds with characters, and of the six available for ”meng”, 蒙 was perhaps considered most appropriate, for the reason you hint (home of grassland). Note that it was the invading Mongol Yuan dynasty who created the Mongol empire from nomadic tribes such as Xianbei and Xiongnu, and the character is if not coined by the dynasty at least propagated and established by it.

The association between Mongolia and Down’s syndrome has nothing to do with character, but is all about similarity in appearance. Chance had likely everything to do with choosing Mongolian features rather than any of the others in the region.

A Mongolian doctor is a barefoot doctor, unskilled people who practiced basic medicine in the early People’s republic, not just in Mongolia but in all of China, althugh they were more common in the more undeveloped regions.

For the same reason, people can use Mongolian as derogatory term, pointing to the more unsophisticated rural habits on the countryside.