a brief
opinion
Oddly enough, a document setting forth the legal arguments, facts and previous court decisions is called a "brief" (legal brief).
A Circuit.
Yes, an appellate brief is an advocacy document presented to an appellate court. It is used to persuade the court to rule in favor of the party submitting the brief by presenting legal arguments, citations to relevant case law, and analysis of the issues at hand.
There is a higher level court than the trial court called the appellate court that hears the arguments. There is usually not testimony during the appeals process. The appellate court is usually made up of several judges and they hear arguments regarding legal issues from the trial. For further information see the related link below.
The type of evidence you are referring to is typically called a "documentary evidence." These documents can include contracts, emails, letters, reports, and other written records that are relevant to the legal case being heard in court. They are used to support the arguments and claims made by the parties involved in the legal proceedings.
The "document" is a court order called a "writ of certiorari."
In the United States federal court system, the document used to initiate a civil lawsuit against a defendant is called a complaint. In the state court systems, this document is usually called a petition.
This time is called the sitting
This is called a brief
In the Supreme Court, the written decision and legal reasoning for a case is called an Opinion.
A majority opinion is the legal document that explains the legal reasoning behind a Supreme Court decision.