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Can I use HD key pairs derived through the BIP32 scheme to derive anything other than secp256k1 addresses? Can I use the same extended private key/extended public keys as are derived through the BIP32 scheme to generate e.g. Ed25519 addresses?

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You can not derive Ed25519 keys in a way that is compatible with bip32 or bip44. Bip32 only defines a derivation scheme for secp256k1, and bip44 depends on bip32. To get around this problem there's a SLIP-10 specification that adapt bip32 in a way that makes it possible to derive keys for other signature schemes than secp256k1. The most important takeaways with slip-10 are that master keys are generated with domain seperation (If one signature scheme is compromised, it wont affect keys generated for another signature scheme). Ed25519 is notable for its public key generation, and therefore the slip-10 scheme restricts it to only hardened derivations.

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