Suppose the House impeaches Trump but the Senate does not convict him and he is not removed from office. Could this House-impeached president run for a second term?
The Senate shall have the sole Power to try all Impeachments...And no Person shall be convicted without the Concurrence of two thirds of the Members present.
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.
The Senate would be judging, so they would have to convict the president of it for there to be a punishment, as that’s how a court works. The judgment is what determines if they would be removed and barred from running again, according to the above quote.
Constitution quote from Wikipedia
Being impeached is the equivalent of an indictment, not a conviction. Bill Clinton and Andrew Johnson were formally accused of wrongdoing, but never actually convicted of anything. The direct text from the Constitution itself:
The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors.
The act of impeachment is distinct from a conviction. There are no official consequences unless the President is convicted.
Yes. Disqualification from holding office is a possible consequence of conviction. Without conviction, there can be no disqualification.
For more information, see Can a US President, after impeachment and removal, be re-elected or re-appointed? at Law.
Yes. Nothing happens directly to the president unless he is impeached and convicted. If he is just impeached he can run for a second term and is a fully functioning president - same as if he wasn't impeached.
It depends. According to the Washington Post, it depends on whether a vote on disqualification from holding office in the future is held:
The question of whether Trump could nonetheless run for president next year is more complicated. In the impeachment of federal officials, the Senate has adopted the practice of holding a separate vote on the issue of disqualification from future federal office after it votes for conviction. Since at least the 1912 impeachment of Judge Robert Archbald, the Senate has required only a majority vote for disqualification.
If no disqualification vote is held, even a convicted official can reenter federal service. U.S. District Judge Alcee L. Hastings was removed from office in 1989 after he was impeached in the House for engaging in a “corrupt conspiracy” — soliciting a $150,000 bribe in a case before him — and convicted in the Senate. But the Senate took no vote on disqualification. In 1992, Hastings ran for and won a seat from Florida in the U.S. House of Representatives, where he remains to this day.
If Trump were convicted by the Senate, but the Senate chose not to hold a disqualification vote, he could in theory run again, win and return to the White House. The path to reelection would also be open if a Senate vote favoring disqualification failed.
Note that article I, section 3, clause 7 of the constitution as quoted in another answer does not state that an impeachment must consider disqualification from holding office, it only states it as being within the Senate's power to issue such punishment.
Politifact also considers the case of separate votes for removing from office and banning one from holding office:
Could the Senate choose what punishment to apply?
Scholars suggest that the Senate has the power to support removal from office without barring them from holding office in the future.
"It is up to the Senate to make that decision," said Stephen M, Griffin, a Tulane University law professor.
There’s even a well-known example of a Phoenix-like rebirth: The Senate removed Alcee Hastings from his judgeship in 1989, but he later won election to the U.S. House, where he continues to serve.
Politifact even goes on to mention precedent for separate votes:
How many votes would be needed?
What’s less clear is the process by which any of this would happen, and the extent to which past precedents would matter.
For instance, "there would be an argument over whether separate votes are required," Griffin said.
However, there is precedent for separate votes, Michael J. Gerhardt, a constitutional law professor at the University of North Carolina, has written.
"In 1862 and 1913, the Senate took separate votes to remove and disqualify judges West Humphreys and Robert Archbald, respectively," Gerhardt wrote. "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."