подрывным студент - disruptive student
This is incorrect. I don't know an analog in Russian, but it may be студент (if it's college/university)/ученик (if it's school) с неподобающим поведением
Студент с неподобающим оведением
Ученик с неподобающим поведением
May be с девиантным поведением.

Ожидания в классе - classroom expectations
I don't understand what is this.

Школьная атмосфера - school atmosphere
Is correct

Поведенческие стимулы - behavioral incentives
It can be used.

любимчик учителя - teacher's pet
Is correct

Учитель дает нам слишком много домашних заданий! - The teacher gives us too much homework!
Is correct
Я не делал этого! - I didn't do it!
Is correct

Она подло! - She is mean!
Incorrect. Подлый/подлая (but no "подло"! Подло is an adverb) = vile. She is mean = Она злая/плохая/дурацкая

In Russian, подлый is sly + evil. It can be used if you describe a man who does something bad to fool. If your "mean" in this case = that, you can use it.

Do schools have principals? Probably they are called something else. What would that be?
Yes, in Russia they are called директор школы. In universities - ректор.

In the USA disruptive students may be sent to the office with a referral and the principal deals with the child. Alternatively, the teacher may call the child's parents and explain the issue and ask for the parents to help motivate the student to comply with classroom behavioral expectations. It's probably the case that similar options are used in Russia.
Yes, there is absolutely same methods in Russia) Only I don't know what is "the office" in this case.

And a question to you: how must I write: Is correct or Is correctly?