I'm aware that when the pronoun is also the object of a sentence we use these: me instead of I, them instead of they and so on.
But when I say:
He is a teacher, and her?
Why do I say her instead of she if in this case I cannot identify a direct object.
Also when answering, would I say
Her, she is also a teacher.
or
She, she is also a teacher.
I'm really confused
I would say "He is a teacher, and she?" (meaning "and what is she?").
I don't think there is any need to repeat the pronoun in the answer. "She is a teacher too" is quite sufficient.
The sentence, "He is a teacher, and her?" isn't grammatical. It contains the fragment, "and her?" In writing, the idea would be better expressed, "He is a teacher. And what about her?" In this example, "her" is the object. In spoken speech, parts of sentences are often omitted but the brain retains the grammar of the complete sentence. This happens often in sentences that point at people or continue action that was started in a previous sentence.
Take, for example, this exchange, which is quite normal;
The volunteer looked around the crowd. "Who do we give the extra money to?" "Him," a woman said, pointing to an injured soldier.
In this exchange, the woman leaves out the idea "We should give the money to..." but speaks the rest of the sentence "him."
So coming back to your example and filling in some of the empty sections, this is how the exchange makes better sense:
The inspector looked at the line of people, then turned to the school principal. He pointed at a man at the end of the line and said "He is a teacher," but then pointed at the woman next to her and asked "And her? (Is she a teacher also?)" "Her?" the school principal responded, nervously. "She is also a teacher."
I can't prove it, but I would guess (and experience with other questions on Stackexchange indicates this) that many situations of "incorrect" pronouns being used are the result of confusion, or different ideas about what parts of dialog are being omitted, and also to the idea of "pointing," which gets transferred to spoken language.
When answering, you want to use the objective form ("Her, she is also..."), otherwise you sound like you are stuttering. In general, I think of it being an ellipsis of a longer phrase, which in this case would be "as for her", which would also explain why it's objective form in the question (but this doesn't entirely explain why subjective form is impossible). The full form ("as for her") is probably more common.
"Her, she..." is a common way while speaking to add emphasis to the subject that follows it ("she"). You don't see it very often in writing, but it's sometimes written differently, such as on its own as a question: "Her? She's also...".
These examples from the internet should help:
See also my answer on ELL that I copied in part to help make this answer:
