It's often cheaper to settle out of court than to go through a court battle. Additionally, and probably more importantly, one does not know the outcome in advance of a trial (either by judge or a jury). Settlement offers the advantage of predictability in that the parties can fashion their own compromise.
Infringement is normally handled in civil court, although the law allows for criminal charges in extreme cases. That being said, the vast majority of infringement cases are settled long before they reach court at all.
criminal
Most civil cases are settled outside of court. Most criminal cases are finalized with a plea bargain.
Civil cases can be "settled" out of court, if the parties are able to reach an agreement. Criminal cases may be plea-bargained if the defendant agrees to the conditions offered by the prosecuting attorney.
Music copyright infringement cases can be brought in civil court, but the vast majority are settled long before they reach that point.
criminal
Even if the case settled prior to trial - if the case was filed with The Clerk of The Court and it appeared on the court's docket, then there would be a publicly available record of it in the Court Clerk's Office. You would have to know where, and in what court system, the case was filed in order to locate the records of the proceedings and learn what the final dispostion was.
court of original jurisdiction
Not really; infringement cases can be brought before any civil court.
Because the vast majority of infringement cases are settled long before they go to court, there is no comprehensive list.
Most states do not accept psychophysiological veracity (PV) examination (polygraph) results in a civil court cases
Yes. Texas has two "supreme courts," although only one carries that name. The Supreme Court of Texas is the highest appellate court for civil and juvenile cases, and the Texas Court of Criminal Appeals is the highest appellate court for criminal cases.