Who handles court cases in the amount over $5,000 (since small claims only goes to $5,000)?

Answer 1

Actually, the small claims limit varies by state.

And once you exceed the small claims limit -- you go to the court of general jurisdiction -- which is usually the general state court.

Most general state trial courts call themselves "superior" courts -- though in NY the trial court is the "supreme court" and then higher courts have different names.

Answer 2

The regular court handles claims above the limit of "small claims court". Usually, this is the "District Court" (unless you are in NY, where they call it the Supreme Court).