No.
They are going under the theory that Barack Hussien Obama was his birth name.
Yes.
Your theory could be right, but TexasDarlin's latest theory is that:
A. Stanley and Obama Sr. got married (in bigamy, while she was preggers already), and that wherever the kid was born, he was given the name Barack Hussein Obama Jr. (or II).
B. Then circa 1966 Stanley married Soetoro and the kid's name was legally changed to Barry Soetoro.
C. That as a teenager the kid decided to go by "Barack Hussein Obama" -- but never followed Hawaiian law for changing his name (even the common law useage law).
D. So "Barry Soetoro" remains his legal name, while he has been running around for years calling himself Barack Obama.
That's her theory, anyway.
http://texasdarlin.wordpress.com/2008/08/12/is-his-legal-name-obama-or-soetoro-act-now/
And from what I've read, in many states, you can legally "establish" a new name for yourself simply by beginning to use that new name [i.e. in many states, you don't have to go through a formal legal name change - you can just start using a new name, and kinda/sorta establish that new name via "common law"].
What's important here is that he LIED to the Illinois State Supreme Court, and the really, REALLY, REALLY important question now is whether he ever travelled on an Indonesian passport after he became an adult [remember, Indonesia did NOT recognize dual-citizenship].