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1. I bet that the Obama camp is scared to death that tech experts like Polariks will one day get their hands on a genuine Obama Hawaii birth certificate, because the Obama camp knows that when people like Polarik finally get their hands on Obama's genuine birth certificate, Obama could end up in deep political trouble.
2. Supreme Court nominee: If Obama is really ineligible to be president, does that mean that he has no legal right to nominate a person for the Supreme Court?
3. That is, is Obama's nominee for Supreme Court invalid if President Obama himself is an invalid, illegal elected President?
Biden has been by Barry’s side for all the major appointments — SCOTUS, cabinet, etc. They have been careful to position things in case Barry is ousted - they will say Biden would have appointed the same people and indeed almost co-appointed the same people... they know what could happen and hope to make things go over smoothly in case it does.
Nominees are confrimed by the Senate, so the answer is no, whomever is nominated is a responsibility of the Senate.