Can padding length, and thus plaintext length, be considered public information when using aes-cbc? Right now I'm using a simple padding system that takes the plaintext, hashes it with SHA512, and appends $x$ bytes of the hash as padding where $x$ is the delta between the plaintext and nearest blocksize multiple.

Is this considered a secure method of padding the message? I am also authenticating the encrypted data using HMAC-SHA512, would this effect the security?
For AES CBC i would suggest you stick to the standards. 

The PKCS7 padding scheme is pretty easy to implement and has been tested extensively now that its been out in the public for a while. I had implemented PKCS7 padding for my AES CBC cipher a few weeks ago and you can see the code here if it helps.

Note that hashing by itself does not provide you with authentication, you will need a MAC (Message Authentication Code) for that. If you like using the SHA family, you can use something like HMAC-SHA256.

Is there any particular reason why you wish to use hashed data as a pad? If you are looking for authenticated encryption, you should stick to the standards like GCM for AES.