"Do you know who the new principal is?"
Translation:你知道新校长是谁吗?
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1224
Without 吗 you would ask for his/her name. This way you ask if the other person even knows who the new principal is (his name, previous jobs and so on).
1076
Without 吗 it's actually a statement of fact, "You know who the new principal is."
To ask the name, you would also need to remove 你知道 from the beginning.
Because it is asking a yes or no question!
Do you know who the new principal is? 你知道新校長是誰嗎 Yes/no 知道/不知道
He is not asking who the new principal is (this would require some kind of description or name), so if he were asking this, it would be 呢), rather he is asking if the person knows who the new principal is. He either knows or he does not know, therefore it is 嗎.
"你們的新校長是誰(呢)?" would be fine, but the nuance is different.
1076
In your first two sentences, you need to move "is" to the end. In those sentences, "who" is a relative pronoun and not an interrogative pronoun, so it doesn't cause an inversion after it.
I think 谁 refers to the teacher (who is the teacher) and 吗 refers to 你 (do"you" know the name). There are two questions in the sentence. I would accept both 谁 and 吗。