[ask] What is the word for the teacher ....

just "teacher"

Generally, the Teacher that acts as a representative for, i.e., "The Class if 2006" (in aggregate) would be referred to as the "Class Advisor".

you teach a subject in a school , so you are a teacher.You are assigned to take care about a specific class affairs and to be responsible about its students like a father , so you are a class teacher.