Suppose Congress successfully impeaches and removes a sitting President of the USA. Can this person still run for a second term?
Assuming the usual rules about term limits are observed, of course.
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Answer: During an impeachment trial, the Senate can "disqualify" an officeholder from holding any public office again, but that is a separate vote from their "removal".
Article 1, Section 3, Clause 7 of the Constitution says (emphasis mine):
Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.
At first glace I assumed this meant that someone who is removed by impeachment is therefore automatically disqualified from holding office, but case law shows this is not how that has been interpreted:
Since ratification, four troublesome questions have arisen under this clause. The first was whether the Senate may impose the sanctions of removal and disqualification separately and, if so, how. The Senate claims that it may impose these sanctions by separate votes: (1) removal, involving the ouster of an official from the office he occupies at the time of his impeachment trial, and (2) disqualification barring the person from ever serving again in the federal government. In 1862 and 1913, the Senate took separate votes to remove and disqualify judges West Humphreys and Robert Archbald, respectively. For each judge, a supermajority first voted to convict followed by a simple majority vote to disqualify. The Senate defended this practice on the ground that the clause mentioning disqualification does not specify the requisite vote for its imposition, although Article II, Section 4, mentions removal as following conviction. The Senate in 1862 and 1913 considered that the supermajority requirement was designed as a safeguard against removal that, once satisfied, did not extend to the separate imposition of disqualification.
So the Senate has the power to vote separately on removal (by supermajority) and/or disqualification (by simple majority), but the one does not imply the other.
You can see this in the current US Senate overview of the impeachment process (PDF):
The Senate may subsequently vote on whether the impeached official shall be disqualified from again holding an office of public trust under the United States. If this option is pursued, a simple majority vote is required.
So yes, an impeached and removed President could still run for office, unless he or she was also explicitly disqualified by the Senate (or was ineligible for other reasons, like term limit exclusions).
EDIT: To be clear, you can be removed without being disqualified, but you can't be disqualified without being convicted/removed.
According to the official PROCEDURE AND GUIDELINES FOR IMPEACHMENT TRIALS IN THE UNITED STATES SENATE:
Following the Vote on Each Article, the Presiding Officer Pronounces the Decision. Once the Judgment of the Senate has Been Pronounced on the Articles of Impeachment, the Trial Might Progress in Two Ways. If the Respondent Was Found Not Guilty on All Charges, the Verdict of Acquittal Was Announced and the Senate Sitting as a Court of Impeachment Adjourned Sine Die. If the Respondent Was Found Guilty of Any of the Charges, the Judgment of Removal and Possible Disqualification From Ever Holding an Office of Trust or Profit Under the United States Was Presented
Impeachment is actually just the first step in the process, and the relevant implications at that stage are political in nature (i.e. reputation, trust, etc.), not legal. Congress approving "Articles of Impeachment" is the equivalent of a Grand Jury finding sufficient cause for a trial — which is obviously quite different from being found guilty. And to date, only two U.S. Presidents have been impeached, and neither one was was convicted by the Senate and removed from office. The relevant ramifications are still political in nature, but it's generally fair to assume that their political career (and probably their personal reputation as well) are dead in the water.
You do raise a very interesting question of whether there are additional legal implications beyond removal from Office if a sitting President is convicted. The process itself is wholly political in nature as the Attorney General’s office has historically viewed it to be unconstitutional to indict a sitting President on criminal charges because it would keep the executive branch from performing its job. Once they leave the White House, they could still face criminal charges, however Presidents tend to have influential friends who can keep them out of trouble by affecting a Presidential Pardon from their successor.
Even in the unlikely event that a President is subsequently convicted of a crime, there's nothing that inherently bars them from running again — unless it is specifically stipulated that they're barred from holding office (either in a Senate ruling or as part of a criminal sentence). The Constitutional framers trusted the democratic process and intentionally left this in the hands of the Senate and the American people.