Is the Chinese character "蒙" being associated with Mongolia and ignorance coincidence?
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.