Sometimes telling us what the plot needs is better than saying what you're doing. Because sometimes we can offer a better way from here to there.
Near as I can tell, you want a situation in a courtroom where one character meets her parents after four years without seeing them, where one is suing the other. But I'm not sure who's suing who here.
However, what you want for a court hearing would seem likely to be a Motion for Summary Judgment. A Motion for Summary Judgment presents the argument that one side, even if all the facts go in their favor, don't have a legal case. For example, Alphonse the Squirrel sues Rocky the Rat for stealing his nuts on 1 April. Rocky responds that not only did he not steal the nuts, he was residing in the State's Hilton Hotel (bracelets provided, but no mints on the pillow) for the period, a fact well known to Alphonse since Alphonse visited him there. Since there is only one version of the facts, (Rocky was locked up which is a perfect defense) there is no way Alphonse can win.
Although a Motion for Summary Judgment is usually filed later in the litigation, it can be filed immediately, based upon the pleadings in the case. The Motion is supported by affidavits (statements made under oath), Filed immediately into the lawsuit, this could very well be the first time the parties meet.
This type of case would never be done in chambers. Either one party or both are at some level nutso, and judges like some protection (like the bench) when bodies start flying around the courtroom.
Now going back to your original question, the courtroom would be open to the public. However, judge's can control the public aspect to maintain decorum. Further, a courthouse in California is used to the problems and knows how to limit the damage. But if your characters are at all famous, yeah, you're going to have some level of a zoo.
Best of luck,
Jim Clark-Dawe