Can I extend my Ed visa at any immigration office in Thailand or do I need to visit the one in my school's province?

NO you can't use any office, you go to the immigration office in the province where you study and your school knows this.

I really hope I do not understand this question. You are telling that you are in a different province that the school where you follow a daily course? The school will provide you with all kind of documents and guide you to the process of the Extension of Stay before your current stays end and this will be done at the Immigration Office in the area of your school.