The first Monday of October is the traditional opening of the Supreme Court. The website teases, "In the event of a lapse of appropriations, the Court will continue to conduct its normal operations through October 4. Further notice will be provided in the event a lapse of appropriations continues beyond October 4."
I confess that I know about the First Monday in October because of the Walter Matthau film. The court calendar is divided into roughly two week intervals. One interval is for hearings and the other is for research and opinion writing.
During a court sitting, each side of the case is permitted a 30 minute argument. There are no witnesses or juries. The justices have before them the complete record of the previous case proceedings in court. Sessions begin promptly at 10 AM, an hour lunch break and then more hearings until 3 PM. Up to 24 cases may be heard during the two week sitting interval. There are no public sessions on Thursdays and Fridays.