Is it cryptographically insecure to use fixed-length AES-GCM messages? Is there any weaknesses to encrypting fixed-length messages? Should a random amount of padding be added to the message to decrease the odds of some sort of attack?
An adversary who obtains AES-GCM ciphertexts but does not know the key cannot obtain any information about the plaintexts except their lengths. This is (in a highly simplified form) the definition of the claim that AES-GCM is secure. This is true for any secure authenticated cipher.
If your plaintexts are intrinsically all the same size, this is an ideal situation, because the ciphertexts leak no information at all about the content of the plaintext.
If your plaintexts can have different sizes and the size of the plaintext leaks some information that you want to keep secret, then padding plaintexts so that they all have the same length is a possible solution. Beware that the generation of the message, the padding process, the unpadding process and the processing of the unpadded messages are likely to take time that depends on the length of the plaintext, so it may still leak through timing. Appending padding to the message may not be the best solution: depending on how the message is structured, it may be easier to construct a secure system if you arrange each part of the message to have a fixed size, and concatenate those fixed-size parts. That way the assembly and parsing of messages is less likely to reveal confidential information through side channels.
Adding a random amount of padding is unlikely to help security. In particular, if the same message is likely to be transmitted many times, an adversary can average out the lengths, which will negate the uncertainty of the random padding. However, it can help in some scenarios where you want to hide which message was transmitted for privacy (for example, to make it harder to figure out which file a user download on a server). Still, the only fully reliable way is to have fixed-size messages (in the download privacy example, this would mean padding all files to the length of the largest one).