I’m no authority on this however I’ve read here on FR several times he can run again if he should be removed from office.
If he’s convicted, the Senate can not only remove him from office but can disqualify him from holding any public office. That’s explicitly stated in the Constitution.
No. Article I, Section 3: "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." The Democrats wrote the impeachment indictment to prevent Trump from ever holding office again if convicted.
But he's not going to removed so it's a moot point.