Not according to US law. If you renounce your citizenship, you sign Documents with the State Department. Or Obama’s mother in this case. He could get it back, but then, because he was a citizen of another Nation, he would be a NATURALIZED citizen. Its not like his mom knew that he would run for POTUS one day. It’s the only time the distinction even matters.
Of course. But there is no evidence that his citizenship was ever renounced by either him or his mother.
And, if he did renounce his citizenship and subsequently became a naturalized citizen, there would be a record of that naturalization. Naturalization requires the filing of an application as well as a naturalization ceremony.