1. Don't make me a fool.
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What does this sentence mean? In which situation can we use this expression?
2. She is the apple of my eye.
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(What are suitable responses to the statement? Could you name them?)
1 means, "Don't make me look foolish." This is a warning of sorts, telling someone to behave and not reflect badly on the speaker.
2 means she is my favorite. Maybe 'she' is my daughter or my granddaughter or just about any female the speaker loves, almost always younger than the speaker.