>What about the symbol Prince used for his name
You don't need to support that. You don't need to cover every single edge case, and that certainly includes people who make up some crazy symbol to be their "name". I'm quite sure Prince didn't file his taxes to the IRS using that symbol, in fact, according to Wikipedia, "1993 also marked the year in which Prince changed his stage name to Prince logo.svg, which was explained as a combination of the symbols for male () and female ()." So that wasn't even his name at all! That was just a stage name. If you're making a system for anything official at all, then you don't have to worry about silly stuff like that.
(On the other hand, if you're making a system for artists which has a field for stage name, then you're going to have to figure out a solution to this one. As you can see with Wikipedia, their solution was to use a SVG file of the symbol and insert that into the text as necessary.)
>What about a language that's not in Unicode yet?
This again probably depends on exactly what your application is. If it's a US government database, then you don't need to worry about stuff like that, because they're not going to care about people using some obscure language that isn't covered by Unicode. Is there even such a language? I doubt it, not any that are in actual use.
Don't forget, just about every language has been Latinized now, so you can always fall back to Latinized characters.
>If someone has a single name, does it go in the first name, or second name spot? Is this enforced, and where?
Irrelevant. Do it however you want, it really doesn't matter.
Now again, as I mentioned before, a lot of this depends on exactly what your application is, so it's pretty hard to come up with any ideas or rules without knowing this. Are you making a database for US government use? Or something for shippers to ship packages around the world? And from what country? Something for payment processors? Based out of what country? Something for the Chinese government? The requirements will change depending on the answers.
If the government forms require a first name and a last name, for instance, and that's the law, and you only have one name, then you have to get yourself a second name. If you have a Chinese name and this is for a US government form, then most likely you have to use a Latinized version of your name. The possibilities are endless.