Question

can the supreme court try cases that haven't been tried in district courts

Answer 1

The Supreme Court hears three types of cases:

2/3 are cases appealed from lower federal courts
1/3 are cases appealed from state supreme courts
Rarely, they hear cases that have not been previously heard by a lower court, such as between one state's government and another.
The justices decide which cases they will hear, about 80 each year. They decide another 50 without hearing arguments. The cases they choose usually address constitutional issues or federal law.

The Supreme Court gets about 7000 requests to hear cases per year, so there are many cases that don't get heard. If they decide not to hear a case, the decision of the lower court stands.