“Obama served two complete terms, so he’s ineligible to be President again”
Nope. He just can’t be ELECTED President again. The Constitution, however, does not even mention Presidential elections.
Nope.
Read the 12th amendment. It doesn't say "elected," it says "eligible."
But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
Either by election or by appointment and succession (like Ford), Obama cannot run for nor hold either office due to ineligibility.
-PJ