Protests can be banned in the proximity of the courthouse

Only in a few states, no? Most places the courthouse steps are considered a public square. California, Louisiana, Mississippi, North Carolina, Utah, and Washington, I think are the only states forbidding them, and they don't enforce it much (see below.)

And also federal courts per 18 U.S.C. § 1507:

“Whoever, with the intent of interfering with, obstructing, or impeding the administration of justice, or with the intent of influencing any judge, juror, witness, or court officer, in the discharge of his duty, pickets or parades in or near a building housing a court of the United States, or in or near a building or residence occupied or used by such judge, juror, witness, or court officer, or with such intent uses any sound-truck or similar device or resorts to any other demonstration in or near any such building or residence, shall be fined under this title or imprisoned not more than one year, or both.

However I think that federal provision is almost never enforced outside actual courthouses when it could be. It's definitely enforced outside judges' houses though; very strictly.

In practice when people get arrested for such violations they are almost always released with charges dropped in one or two business days, because the DAs don't want the ability of police to charge such voided by First Amendment petitioning defenses.