1. This is the Supreme Court of the USA.

Question

2. These are the Supreme Courts of the USA.
3. This is the supreme court of the USA.

(Which one is correct? Do I have to capitalize? Do you have only one supreme court? Or do you have many supreme courts?)

4. the Supreme Court > a high court > a district court

(Are they correct?)

Answer 1

#1 is correct. There is only one for the country.

I don't understand what you're signifying in #4. Please explain.