Joseph,
1. An American trial judge is more like a referee than a judge. The jury decides the verdict. the judge only relates the legal issues after receiving opinions by the two lawyers. An appeals judge can use Talmudic reasoning and psychological insights. They can also cite Jewish law to support their opinions. CJ Warren cited Rambam’s explanation of why a confession is inadmissible in a criminal case in the “Miranda” decision and Justice Scalia cited Chazal’s statement that a judge who accepts a bribe loses divine inspiration in “Caperton v. A. T. Massey Coal Co” (see “Justice Scalia As Talmudic Scholar” by Natahhn Lewin on-line).
2. Did you read Rav Moshe’s teshuva? I did, however, hear about a case where two Chassidim took their case to a secular court and the frum judge rebuked them for not going to a bet din. I do not know about the US but in Israel judges can clear their calendars by sending the parties to arbitration or mediation. The bet din of Eretz Hemdah-Gazit has such a sterling reputation that even non-observant judges refer cases to them.