I concur with you. Still he did have the right to have her remove herself from the court. As far as the defendants go, I can’t be sure what might be the legal reality there.
Just because a judge asks you a question does not mean they have the authority to ask it or that you have to answer it.
His question had nothing to do with ongoing court cases either. Its a personal medical status question of someone. Asked in a public venue, no privacy whatsoever, anyone can hear it, and the courtroom may be recording at that time.
Besides one could answer “cucumber!”. Might not like the answer but theres no law that says he has to like your answer.
Or answer with a question of your own. “Have you been vaccinated?” “Whats the length of your erection?”
She had no choice. They are federal prisoners. She couldn't leave them with the county or state officers. From the time she picks them up from some federal location until she brings them back to that federal location, they are her responsibility. She may have had to cross state lines in moving them. That is the US Marshall's purpose. Crossing state lines which state/county LEOs can't do.