1. He works at the Supreme Court.

Question

2. He works at the supreme court.

3. He is a judge of the High Court.
4. He is a judge of the high court.

5. He is a judge of the District Court.
6. He is a judge of the district court.

7. There are several District Courts here.
8. There are several district courts here.
(Do we have to use capital letters or small letters?)

Answer 1

http://grammar.ccc.commnet.edu/grammar/capitals.htm

I would use #1, because to me it automatically implies the US Supreme Court. But if it's referring to another country's high court or supreme court, I'd leave the capitals off unless I included the specific name of the court. (The High Court of Australia, for example.)

In 5 and 6, again, the capitals would be needed if the specific court is named, such as Harris County District Courts.

I wouldn't use 7 unless the specific name is used. I'd use 8 if the specific name isn't used.