“Qualified” in the 12th Amendment refers to receiving the requisite number of Electoral College votes to win.
In Obama’s case, he “qualified” when Vice President Cheney certified the Electoral College vote at the Joint Session of Congress with no objections from any one Senator and any one Representative.
http://legal-dictionary.thefreedictionary.com/Twelfth+Amendment
There is no constitutional or legislative process to remove from office ex post facto, a person who has qualified via the electoral college and been sworn in by the Chief Justice other than with a Bill of Impeachment passing the House and a trial and conviction for High Crimes or Misdemeanors in the Senate.
The term "President elect" is not a legal term until such a time that the Congress authenticates the results of the electoral college votes. Your twelth Amendmenment "qualification" has already taken place when the Twentieth amendment "qualification" kicks in.
In addition, there are no electoral college votes involving whomever the Congress declares "who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified". Since Congress is empowered to select an interim President, the electoral college votes may or may not come into play, thus, there is a different sort of "qualification" required here.