This is America, the melting pot. We are individuals (people/groups), but we are one. In order for that to make any sense, we have to be recognized as what we are. Just from a social standpoint, to name everyone American doesn't make sense to me. All of the individual cultures that make up America have so many interesting and wondrous attributes. I'd never want any of it muted.
Doing that to it mutes its integrity. Foreign people migrated/migrate here and it is still known as the land of promise, but they still love and draw inspiration from their homelands. Most left because of injustice, but that doesn't mean they don't want to be what they are anymore. Being American should never be about not being who you are...whoever/whatever that is. It's because of America's policies that people are free to be/ identify, claim the name of whatever they want. You talk about what "American" is, being able to recognize your roots/culture/race/identification is the most American thing I can think of.
As far as why is race asked for on certain documents like applications and such, well, it's for identification purposes. You know, identification: what if you were working at a place and the FBI wanted to question a dangerous criminal who just so happened to be working along side you? How far do you expect them to get asking: "I'm looking for an American man, he has American skin, American eyes, and American hair?"
Race should be a source of pride. As humans, we do thrive from a little healthy competition and affiliation. That Native American scholarship is for Native American students who have excelled, an African American scholarship, like wise. Why exactly is having your affiliation recognized, offensive?
African American: this is a term I love. For one thing, it's better than being called black and for another, it does something many of my race fail to do: recognize their roots, recognize their affiliation. You say duel citizenship, well it sorta is. We have unfinished business in Africa. We didn't leave willingly and our children were not given a choice, to either identify with Africa, America, or both. For me, the term symbolizes that ability to choose, which didn't exist in the past, every time I hear it. Being born American doesn't, for me, erase history, erase ancestry, origins, nothing. In a way, the whole situation is demonstrative of the America you think you want: black children being taught the history and culture of other people. Yeah, I know there is a lesson on Martin Luther King during Black History Month every year, but that doesn't even scratch the surface of our culture or identity. That's history. For those of you who went to a good school, congratulations, but for most black people, self education is just about the only way they'll learn about Africa, which is where their ancestry probably is.
Anyway, you are biracial; you should be having a field day exploring and dissecting both facets of what and who you are. Your race has a LOT to do with who you are, in ways, shaped it. You live in America, but you aren't American because America is made up of too many things. The term "American" is just for identification purposes, I think, anyway. You’re American because you live in America now who are you?