If he's proven ineligible (e.g., if the SCOTUS rules that he's not a natural-born citizen according to the law), would impeachment even be in order?
You impeach the President because he has committed "high crimes and misdemeanors," according to the Constitution. But if the person in the WH is ruled ineligible, then he's not the President and never has been. He's a private citizen (or maybe not even that) occupying government property in violation of law.
The Constitution is very clear on this matter. The Supreme Court does not have the Constitutional authority to remove a sitting President (and may not even have oversight to determine eligibility). Once a President assumes power, only an impeachment and conviction can remove him.
As we have seen, the Courts are very subject to the political whims of a single individual, and the public has no oversight of the Courts. The Founders likely did not want to leave the authority to remove a President up to such whims.
Thus, only the legislature can remove him. Assuming office fraudulently would certainly fall within the purview of a “high crime and misdemeanor”.
SnakeDoc