What is the ruling on my taking the answer to an exam question from the teacher who is grading my paper? And must I compensate for that by having grades deducted from a subsequent exam? And what should I do if the teacher inadvertently counts a wrong answer as correct, and must I also compensate for that?

What the teacher did by giving you the answer is not permissible for him, and he bears the sin for it if he was not authorized to do so. If the teacher made a mistake in the correction, you must refer back to him to adjust the result. If you are unable to do so, there is no blame upon you, and you are not required to intentionally get a wrong answer in another exam. The forbidden acts are cheating and forgery, and you did not do any of that.