Created a library for the MDFLY MP3 module (TDB380)

Yes. I have never experienced that issue. Is it possible that your MP3 is corrupted? What's especially odd is nearly every behavior you have described just keeps coming back to a GND not connected between the Arduino and the module. Unless you are 100% certain, I would suggest measuring this connection to make sure it is not broken.

The not being declared part. Are you spelling it correctly?