All the impeachments in the world don’t prevent him from running again. A felony conviction would be required for that.
Article 1 Section 3 says that part of the punishment if convicted is a ban from holding any office in the future. “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.”
Other way around. Nothing in the Constitution prevents a convicted felon from running for president. But an impeachment conviction prevents him from ever holding office again.