Well, lets just say he must be legally assumed to be the President. However, he either is or is not eligible to the Office of President. If he is not, he's not the President. Somewhat like an accused murderer. He either is or is not a murderer, but for most puroposes the law assumes he is not. Until the trial that is.
In this case it might be until the discovery phase of a trial, or the trial itself. It could even be the fact finding phase of Congressional hearings (I would not hold my breath waiting in the case of this Congress).
However once it was completely proven that he was not eligible, impeachment would not be an option, as only "the President, Vice President and all civil Officers of the United States"may be impeached", and then only for "Treason, Bribery, or other high Crimes and Misdemeanors.". Fraud to obtain the office of President would surely be a "high crime", but if he's not President, he doesn't come under the jurisdiction of the House.
The only process to remove a President once he has been sworn in, prior to the next election, is impeachment and removal.
Courts could rule on eligibility in the context of a ballot challenge prior to the election, ie, an attempt to kick an ineligible candidate off the ballot. But after he is sworn in, no court process could kick him out of office except impeachment. As noted, even a felon could still serve as President, until impeached/removed.
I would appreciate it if you would link to the law that governs the above.