I don’t think there’s any Halachic issue with biting nails.
You should throw them out, burn them, or flush them down the toilet (some have a preference to burn them but there’s probably no actual difference). Don’t ever leave them lying around.
The Gemara says that a woman who walks over a fingernail can lose her baby. I once heard a wonderful P’shat that it probably means that if she steps on it she might think she was just bitten by a scorpion and the fear/trauma will cause her to miscarry. But we assume that it’s a Kabbalistic Inyan and therefore you should never leave them lying around, even though scorpions are not present around us nowadays in most places.