Does having a hash of the plaintext compromise the ciphertext? If I have a SHA512 hash of my plaintext, does it weaken or break the encryption of my ciphertext?
Yes, having a hash of the plaintext available weakens the security of your encryption scheme.

Semantically secure encryption schemes have the guarantee that ciphertexts should be indistinguishible, regardless of what plaintext is encrypted. Clearly hashes do not have this property, since they're deterministic for each plaintext.

Assuming that messages are not randomly chosen, an adversary can guess messages, hash them, and check whether or not the hash is correct in order to learn the plaintext for the ciphertext. With only access to the ciphertext, but not a hash of the plaintext, this would be impossible, since a semantically secure encryption scheme does not allow an attacker to guess the plaintext for a ciphertext, as ciphertexts appear to be completely random with respect to the plaintext. The confidentiality guarantee of your system should only rely on the randomness and size of the key used, and not on the randomness of the plaintext, so this scheme is insecure if a hash is provided.

As for whether or not it allows the attacker to generate the key, that depends on the security guarantees of your encryption. As long as the encryption scheme is CPA secure (which is pretty much the weakest level of security that's "acceptable" for an encryption scheme), then one can't recover the key from plaintext and ciphertext.