Problem is no precedent for quo warranto against sitting US president in any district court and the DC court is the one with the best prospects, not the Third or any other non-DC circuit court. Judge Carter said as much. The Third Circuit can easily punt the quo wararnto part of this case, and I expect that they will.
Regarding failure to verify original BC and NBC status, I don't see a remedy available to the courts under separation of powers, now that Obama has been inaugurated.
Where is that written in the Constitution?
Inauguration isn't even explictly mentioned in the Constitution. Just a requirement to take the oath of office, along with being eligible, 35+ 14+ years resident and Natural Born Citizen, get majority of electoral vote. All requirements. Meet them all, you are President, even if you take the oath by signing a form filled in with your name. No requirement that the oath be oral, on bible, by Chief Justice, etc, etc. Inaguration is traditional not Constitutional.
When the Chief Justice does administer the oath, he is not acting under his Contitutional responsibilities. At least I don't see anything about doing that in Article III of the Constitution.