Yes, an impeached US president may still run for election. Impeachment does not bar an impeached official from running again.
However, if Nancy Pelosi who is requesting for the filing of a second impeachment against Trump in the light of the violence that happened at Capitol Hill, makes good her threat if Vice President Mike Pence does not invoke the 25th amendment, this, I think, would lead to the end of a Donald Trump. This is better than the humiliating and ceremonial kicking his butt out of the White House. Humiliation will not stop a Trump from resuming power ever again as President.
Should the House do pass and thru a summary hearing, find Trump liable of committing sedition or insurrection, impeachable offenses, then that resolution may actually contain a clause meting Trump a permanent disqualification to seek or run for an elective office during his lifetime.
After his impeachment, government may run after him and continue prosecuting him with sedition or insurrection or even coup d’etat. A guilty Trump may not just go straight to jail– he will be meted a perpetual disqualification from election into public office as an additional penalty aside from incarceration.
This route, I think, would permanently end Trump’s political career and would just make him a poster boy of a failed cause.