It is widely accepted O was adopted by Lolo Soetoro—hence O’s name of Obama ( Barry ) Soetoro on school records which have been shown on this forum.
As to the other point, I understand that Indonesia did not allow dual citizenship—i may be wrong, show me the refernce on this—in which case the citizenship of the father ( adoptive) is passed to the son.
Not under U.S. law-- if you were born a U.S. citizen, you can never lose your citizenship unless you personally renounce it, as an adult, before a U.S. consular officer overseas. (See the U.S. Supreme Court's decision in Afroyim v. Rusk). Your parents could renounce your citizenship 100 times when you were a child and it would have no effect.
So you will accept a Un-vetted photograph that says Obama's was born in Honolulu, Hawaii.