yes and yes
IF he was born overseas, she was not 5 years older than her 14th birthday so he is an Alien that was the law in effect.
IF he was born in Hawaii, its possible she renounced his US citizenship by taking him to Indonesia where he was legally adopted.
He could not be a secret service agent or even own a gun if he renounced his citizenship via his mother.
No, a parent cannot renounce citizenship on behalf of a minor child. Only renunciations of citizenship by the person after attaining the age of 18 are valid under US law.
The only question open is whether he was a citizen at birth, which because his father was a British subject, and his US citizen mother was younger than the then prevailing statutory requirement for one parent’s US citizenship to be passed to a child at birth, turns entirely on where Obama was born.