A course can only be "run" by a teacher or teaching organisation, not by a pupil/student. In your example, "Do you run an english course at your school?"" means "Does your school run/offer an English course?".
"do" or "take" would normally only apply to the pupil/student, but "do" could also possibly (just about) be used in place of "run/offer" by a teacher or teaching organisation, but this usage would depend on the context being clearly established I think, and would be a little informal.
To run an English course would be to be the head of an English department. It is awkward sounding, as you would rather run the English department than run an English course. Do an English course is also awkward--although it would be understood. You would TAKE an English course.