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i guess it must be 'aide'

Завуч = заведующий учебной частью, lit. teacher responsible for study and learning section (higher authority than just teacher).
Зауч - might be some student slang, but this spelling is incorrect,...

Classroom rules - broadly used term is правила поведения в школе (lit. school behaviour rules, implying classroom rules too), or правила поведения в классе - this makes sense too. Hence classroom...