Is running the red light considered a crime and will it show in US embassy or any other background checks?

Minor traffic violations are usually not considered criminal and failure to stop is a minor traffic violation. It shows up on your driver's record but I have never seen it reported anyplace else. The reason for Traffic Violations as opposed to Criminal Traffic Violations is the Constitution, usually. Since you are entitled to an attorney if charged with a crime, most States don't want to pay for such a practice nor do they want you to be able to have a jury trial for a minor issue like failing to stop at a sign or signal.