Your are correct there! Elena Kagan has been serving as the Solicitor at the Supreme Court, and she is the one who decides which cases go forward to be heard by the Justices.
BTW, she also was Dean of Harvard Law. Funny how she finds herself in the perfect jobs to be able to protect Obama.
No wonder he nominated her for the Supreme Court.
No, only cases where the government is the plaintiff. Any one can appeal a case to the Supreme Court, they don't even have to be a lawyer. The court gets thousands of appeals every year and each and every one of them gets reviewed by one of the justices, almost always in the form of a synopsis prepared by one of their law clerks. Once a week all the justices meet and vote on these cases and decide which ones will me heard by the full court and which will not. So the fact that this case has made it this far to the Supreme Court is not surprising. As I said other cases have as well. But the court has denied all the Birther cases that they have reviewed so far, and I doubt this one will be picked up either.