Does US law prevent (or frown upon) defendant from publicly publishing evidence, even when the plaintiff publicly published the accusation and their evidence first? If so, how is the defendant supposed to protect their public reputation when a lawsuit is ongoing for years? Or maybe I’m missing your point on what wouldn’t be looked upon kindly?

Two wrongs don't make a right, even if the plaintiff is playing dirty it doesn't give you the right to join in.

Assuming the exhibits aren't sealed they're publicly available as soon as they're entered by the court during trial.

If they're sealed there's probably a good reason they shouldn't be published until the trial is over (or ever in some cases)