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The only requirements for a President are spelled out in Article II, Section 1 of the Constitution and Article I, Section 9, Clause 8 of the Constitution shown below, respectively.
"No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty five years, and been fourteen Years a resident within the United States."
"No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State".
That's it with respect to the Constitution. Those words have been subjected to interpretation by Congress and the courts.
The Ethics in Government Act of 1978 passed after Nixon's removal from office, created mandatory, public disclosure of financial and employment history of public officials and their immediate families and created restrictions on lobbying efforts by public officials for a set period after leaving public office. It also created the U.S. Office of Independent Counsel, tasked with investigating government officials.
For compliance with the Ethics in Government Act, the Office of Government Ethics (OGE) is an independent agency within the executive branch of the responsible for directing executive branch policies relating to the prevention of conflicts of interest on the part of Federal executive branch officers and employees. Walter M. Shaub Jr. was the head of the agency when Trump took office, but resigned in frustration with (as per NYT article) the "way the Trump administration had flouted or directly challenged long-accepted norms in a way that threatened to undermine the United States’ ethical standards." Shaub upon his departure said, “I think we are pretty close to a laughingstock at this point.”
Rather than following the normal line of succession, Donald Trump appointed David J. Apol, to take over the head of the office. Apol was chosen by Trump because of his known stance on seeking to "roll back or loosen ethics requirements on federal employees, including those in the White House". In other words, Trump found a friend in Apol in so far as any investigation of Trump's ethics in office is concerned.
In compliance with the Ethics in Government Act, in May 2016, Trump filed a 104 page financial disclosure report of his financial holdings around the world, but these excluded his income tax returns which would have provided critical insight into how much Trump has benefited. Likewise his 98 page June 2017 disclosure was similar in lacking in details.
As per CNN Money, Norman Eisen, a Brookings Institution visiting fellow and former ethics lawyer for President Barack Obama, said with respect to the 2017 disclosure, the document is missing a great deal of valuable information. "We still don't know the extent or sources of foreign emoluments, the identity of all his investors, partners and financial actors involved in his businesses, the purchasers, including possibly foreign ones of his condos and other properties."
Slowly the truth will be revealed unless somehow Trump is successful is shutting down or limiting the investigation by Robert Mueller. That right now is seemingly our real path to the truth as Apol is proving to be yet another Trump enabler.