He was a Kenyan national at birth through his father. Do you have any cites that he has renounced his citizenship? If not, then referring to him as Kenyan is accurate.
It was his step-grandmother. And it wasn't quite immediately corrected.
Per his bio (which I believe are approved by the author of the book) in a promotional brochure for his first book, it describes him as being born in Kenya.
I don't get into all the birther stuff. Whether or not he was born in Kenya, I've considered him a citizen by virtue of his mother's citizenship.
But I've always wondered the reason Obama refused to release school records. Did he apply as a foreign student to get bumped up on the list? It was my understanding that he wasn't exactly a terrific student. How did he get in? I think the press largely dropped the ball in vetting Obama in 2008 because they were utterly smitten by him. I don't know that if we saw his school records we'd find anything to really care about. But since there is the question, they should have pushed for an answer as fervently as they pushed for Romney's tax records 4 years later. I get the feeling the press didn't push for it because they were afraid they'd have to report what was in it.