Nevertheless, since there is a factor mod n at the end, I couldn't tell wether Kchild of Case 1 will relate to that of Case 2.
Yes, the child private key from case 1 is the secret key for the public key derived in case 2. $n$ is the order of the generator point $G$ so it doesn't affect the correctness of the secret key.
since HMAC function takes in 2 input which are key and text, what is the "key" in this case? If there is no "key", then why not using just SHA-512 hashing function?
The key is the "chain code", while the text/data is as you wrote above in your question. This chain code ensures child keys are not solely derived from the parent keys, but rather also use some extra entropy which must be known in order to derive the children. So you can choose to give out the chain code if you want people to be able to derive child keys, or keep it secret so they cannot.