First, Judge Judy is not a real court. Judith Sheindlin was a real judge, but then she retired to be on TV. The show depicts arbitration proceedings, which both parties agreed to. If both parties want it, and if he agrees, you can have arbitration proceedings where R. Lee Ermey presides and keeps order by means of screaming at you and ordering calisthenics on the spot (where failure to comply means you lose automatically). Judge Judy is not a public servant on the show; she's a TV personality.
That said, real judges have substantial power to keep order in the courtroom and to enforce proper decorum. Part of proper courtroom decorum is that you respect the authority of the court, and that you respect the judge. Failure to do so can be criminal contempt of court. Contempt of court in the presence of the judge is the only situation I'm aware of in American law where you can do something wrong and be pronounced guilty of a crime and sentenced to jail on the spot without the ability to even say anything in your own defense beforehand (some places require opportunity to be heard before sentence is imposed or the order becomes final, but not all do). Part of the point of this is that courts only have authority because people are willing to obey them; one of the actual reasons for summary direct criminal contempt is to vindicate the authority of the court. If you're acting like the court can't tell you what to do, the court can send you to jail to make the point that they absolutely can tell you what to do (in practice, jail sentences are often commuted when the contemnor apologizes).
With the "Adios" thing, that wasn't a fine: it was bail. Bail is the amount you pay to guarantee you'll show up in court later (if you show up you get it back; if you don't the court keeps it). Disrespecting the court means they're less convinced you will show up because of a sense of obligation, so they can increase bail to add financial incentive.
In his personal capacity, a judge can be criticized like anyone else. In his courtroom, a judge is not the peer of anyone else in the room. The judge is the court; he represents the law (judges used to be stand-ins for the King; now that the US is based on the rule of law rather than the rule of a King, judges are stand-ins for the law). A phrase sometimes used is that you may or may not respect the judge, but you respect the robe.
So, yes. If you think a judge is disrespecting you, you are expected to take it and not show disrespect right back. You are expected to show respect to the judge, to the proceedings, and to the other people in the courtroom. Judges occasionally abuse this; there are stories of "black robe fever," in which a judge goes off on someone in the courtroom for no good reason. But that doesn't excuse anyone else in the room from their duty to show respect to the judge while in court.