That would have to ignore the fact that many of those actions taken unilaterally by the Executive Branch per his orders were beyond the limits of Constitutional power. Those edicts, appointments, etc. could well be void because there is no way to make the case that they were done with the approval of the Legislative Branch, something which could be done with legislation passed first by the Congress.
You're right that a number of actions taken by Obama are ultra vires and should be considered invalid by a court.
But this is true whether or not he is ineligible for office.
For example, Obama does not have the authority to exempt an entire category of people from the immigration laws, as he has done with his decree as to non-enforcement of the law against illegal aliens who fall within a certain category.
This action is by nature a legislative act and beyond the power of the president and should be declared by an honest court to be invalid.