The past form is used there since she is actually asking about his past experience "whether he saw/have seen her essay or not", not the past existence of her essay.
The present form is ありませんか?, not ありませんですか?.
ありませんか? is also valid. She is asking about the existence of her essay directly in this case.
I had supposed she was referring to the piece of paper he was reading. So it was right there when she asked so past tense should not be used. Thanks for the suggestion of alternative thinking.
Her words work perfectly fine even if what he is reading is her essay. She might not really realize it, or she actually already realized it, and intentionally asked so (in a polite manner, or maybe in a sarcastic tone).
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