In Korean, the family name comes first, then the given name. (There are no middle names.) It is almost always a single-character surname (Kim, Lee, etc.) and a two-character given name, though there are exceptions, with a small minority of surnames being two characters, and a small minority of given names being one character.
Romanization rules have changed over the years. The first president of the modern Republic of Korea was Syngman Rhee, with his given name Syngman first (with no spaces or hyphens) and family name Rhee last. Then take Park Chung Hee, the third president, in office in the 60s and 70s; his family name is Park, and his given name is Chung Hee. A few presidents later, you have Roh Tae-woo, in office around the 90s, with family name Roh and given name Tae-woo; notice the hyphenation. The current president is Yoon Suk Yeul, back to not having a hyphen to join the two characters of his given name.
Sometimes it depends on when it was that someone was first issued a passport, because the government is loathe to change the way someone's romanized name is spelled after the first issuance.
I was confused for a long time when I saw the name of Namkoong Min, I thought they wrote it reversed for some reason. Now that he's in the news again, I realized Namkoong 남궁 is the family name -- never seen one like this before.
Also, in (South) Korea, depending upon the person's preference, they may choose to do family name first or last. It's crazy. I don't know why. (Can someone explain it to me?) As a result, most will write their romanized family name in all uppercase. Then, it can easily be identified as first or last. The easiest place to see it is in movie credits. When the screen is full of romanized Korean names, you see a mix of family name in all upper as first or last.
South Indian names can be incredibly long. Many people will shorten to use a single character from either first or last name.
Just have a single free text field. That should work for all cultures. Allow for 64 chars minimum (hello Spaniards!) and possibly 128 chars. Also, you might try a second field for "nick name". Lots of people with very long names use a shorter form in less formal settings.