The Noonien-Singh Institute was probably not named after Khan himself, if it's where he lives. Instead, it might be the last names of one or two people who founded it. If it's two people, then perhaps his mother and father; one from India and the other from China.
I used to think that Noonien was a Chinese name -- since Roddenberry allegedly named Khan and Noonien Soong after a Chinese person he was trying to get the attention of -- but I was later told that it is indeed a name used among Sikhs. Although I've never been able to find any references to the name outside of Trek, so I'm not sure what it actually is.
In which case, any children produced in the program could also have been given the surname due to their origins in the program, not just those whose biological mother and father were members of a family by that name.
Well, yes, I think that was the obvious implication in "Tomorrow and Tomorrow and Tomorrow," that Khan was named for the institute.
To add to the above....Khan is meant to be related to Genghis Khan
Where the hell did you get that idea? That has never been suggested in any canonical work. It can only be a fan theory, and it's a stupid one, because Genghis Khan's actual name was Temujin. Genghis is an honorific, and "Khan" is the title of any and every Mongol chieftain.
Also, Khan is an extremely common surname in Asia and the Muslim world, because of its history as a title, so assuming that someone named Khan was meant to be related to any one specific khan is like assuming that Alan King was named after King Arthur.
Also it makes sense since Genghis Khan was know for his brutality, as was Khan Noonien-Singh.
Both these statements are wrong. Genghis is painted by Western history as nothing more than a violent brute, but while he was indeed brutal to his enemies, so was Alexander the Great to an equal extent, and both Genghis and Alexander were equally tolerant and benevolent to those who submitted to their rule. It's only due to racial bias that Western history teaches Alexander as a hero and Genghis as a monster. Asian history paints Genghis as a revered cultural founder in much the same way Western history paints Alexander.
And "Space Seed" explicitly stated that Khan was
not known for his brutality. "He was the best of the tyrants and the most dangerous. "There were no massacres under his rule." "No wars until he was attacked."
Christina Chong was not meant to be a totally different ethnic group, but a subset of the Khan Noonien-Singh ethnic mix, and purposely so.
You do me a disservice by misusing my quote to support a premise I do not agree with. I was absolutely not saying that any producers of
Star Trek actually intended Khan to be ethnically mixed. The passage you quote was in reference to the casting of Benedict Cumberbatch as Khan; my point was that,
hypothetically, there was no reason that Khan could not be a different ethnic group from the South Asian one presumed by "Space Seed." I was not saying he actually
was, just that he conceivably could be.
However, to play the young Khan in "Tomorrow...," the producers of
Strange New Worlds cast Desmond Sivan, a Latino actor who, depending on the source, is partly South Asian or at least appears South Asian. So they were splitting the difference between Montalban's real ethnicity and Khan's intended ethnicity. There is no suggestion whatsoever that they intended anything East Asian in the mix.
And there's no reason why they should. After all, Khan ruled a quarter of the world from India to the Pacific. He probably had concubines from all over the region he ruled, so he probably has descendants of multiple Asian and Pacific ethnicities. After 7-8 generations, as I said, his own genes would be diluted to the point of near-indetectability, so he could be fully Desi yet still have descendants who showed no trace of South Asian ancestry. Thus, there's no reason why Khan would need to have any East Asian genetics in order to have East Asian descendants.
Right now they are flushing out her experiences with the Gorn.
It's "fleshing out." It means building up, expanding. "Flushing out" implies the opposite.