Legal status of the judges outside of the courtroom: Rosh HaShanah 30a
Hearing the Litigants
Making a claim on behalf of a woman to justify her receiving the ketubah money despite lack of a hymen, where she didn't know to make that claim herself, as a function of the principle that judges should "open their mouths for those who are mute": Ketuvot 36a
A judge may not be gentle to one litigant and harsh to another: Ketuvot 46a
If a judge does not speak up when one litigant threatens the other with a legal consequence, onlookers may assume the judge considers the threat legitimate: Ketuvot 69a [and see Tosafot ואישתיק]
The Verdict
What if there is a 2-0 decision, with one abstention: Sanhedrin 5b
What happens if a party flees following the verdict, and then returns to a court: Makkot 7a [2x]
A party who fled following a verdict outside Israel, to Israel: Makkot 7a