No. Only the passport on which you obtained a green card will (might) be of interest during the N-400 interview. The other one is irrelevant.
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Why wouldn't you want to show it to them? If it has prior US entry stamps, I'd bring it along.
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It does not hurt to bring it along to the interview.
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Bring it and present it if asked. The immigration authorities see a lot of passports with what looks like damage, but actually have been tampered. I think an experienced immigration officer would be skeptical about a rationale for not bringing an old passport based on it being damaged. Here in the U.S. there is a joke about a child explaining why he did not bring his homework to class "because the dog ate it." Better you let them see it if they ask for it.