Achaia was the Roman province in which Corinth was located. Corinth was the largest city in southern Greece and the capital of the province. (See: [[rc://en/ta/man/jit/translate-names]])
This phrase means that the Jews began to act, not that they rose into the air or rose to stand from a sitting position. If your language has a different idiom for beginning to act, you may use it here. (See: [[rc://en/ta/man/jit/idiom]])
The Jews took Paul by force to bring Paul before the court. Here "judgment seat" refers to the place where Gallio sat when he made legal decisions in court. Alternate translation: "took him so that the governor could judge him at the judgment seat" (See: [[rc://en/ta/man/jit/figs-metonymy]])