- I think you're right that the box is supposed to mean native peoples whose nations currently are inside of the USA.
- It's possible to note your indigenous status in the southern hemisphere elsewhere on your application. Of if you don't find a place, it's generally okay to email the admissions office with one or two lines that state your identity.
- I don't think they require proof of your ethnic heritage in the US. Since the 2010 census, if my memory serves me right, officially US denizens were given the option of marking how they identify and could mark more than one box, if that was true for them. I'm sure that the colleges would accept that official standard too.
- "native American" and "American Indian" are largely interchangeable words, although some people take offense at "American Indian." I've also heard some take offense at both terms, because a person might not be Native American or American Indian; he or she would identify as Hopi or Sioux or Shoshone or one of the other tribes. Those nations existed before there was an America and they still exist. Some people's ancestral tribes were destroyed and so have no official tribal affiliation, though they have native ancestry. They also do not have "proof". To my mind, that doesn't make them any less native in origin.