First off, "fails to qualify" means that Congress must name an interim President and if this is the case, the one from the election cycle didn't "qualify". The Amendment talks about the President Elect either dying before taking the oath or "failing to qualify". In both of these circumstances, we are talking about a "President Elect", a person who has been identified as such does not officially exist until all of the election is over and Congress has banged it's gavel accepting the results. This certainly means that this "qualify" business is something OTHER than election results. What could it be other than the qualifications to hold the office of President, the eligibility requirements from Article Two, section Two?
In order to prove eligibility, being 35 years old or older and being a natural born citizen, it would seem a birth certificate would do the trick. The term "or if the President Elect shall have failed to qualify" describes an action of the President Elect that he failed at so clearly it is the burden of the one seeking the office to qualify.
The 20th amendment does not say that if the president-elect fails to qualify congress must select a new president. It does say that if a president-elect and the vice-president-elect fail to qualify then congress may pass a law describing how a new president is selected. The Presidential Succession Act of 1947 already lists who becomes president after the vice-president.
If Obama were to be declared ineligible then Biden would become president under the 20thAmendment.
Who decides what is qualifying evidence? Congress?
While a birth certificate would do the trick, it is not required. Congress could decide to use whatever evidence they deemed sufficient. For example the certifications sent to the individual states made by candidates after the conventions might be considered as qualifying evidence.