Can high school teachers teach the same subject but different grade levels?

Depends on the subject and school. At most schools, world history is taken by sophomores. They could teach U.S. History to juniors as well but that would be a different kind of history. The only subject a teacher can teach where they will usually have multiple students in more than one grade level is math. For example, in a class like geometry, there is usually both freshmen and sophomores.

It depends on how the state teaching license is written. Typically, a math teacher could be certified to teach math in grades 9-12 or even 7-12. That means a math teacher could be assigned to teach algebra 1 or trig or any math course. Of course, some may have preferences, but they have to teach the classes they are assigned (with their area of certification).

Yes. If I am understanding your question correctly, you can teach the same subject in different levels. You can teach math one period to freshman then the next period have all seniors. It just depends on the topics that will change though.

Most of them have to. Very few schools have 6 periods of sophomore anything.

Of course.