Sam Levine at the Guardian explains the “is he barred from running?” question this way:

Can Trump still run for president?

Yes. The US constitution does not prohibit anyone charged with a crime, nor anyone convicted of one, from holding office.

The 14th amendment, however, does bar anyone who has taken an oath to protect the United States and engaged in “insurrection or rebellion” from holding office.

Relying on that provision, a slew of separate civil lawsuits in state courts are expected in the near future to try to bar Trump from holding office.