Any luck with disabling sleep?
If you're able to disable sleep and make the problem go away, then we know it is sleep related. I don't use my Mini as a MC Server (it is a client of my Windows machine), so I don't have any direct experience with what you're trying to accomplish.
Maybe there's a bug in how MC handles waking a Mac with a USB audio device? Or, maybe, OSX's Power Management API just doesn't allow it to work (or is buggy itself)? Or, maybe your audio device's driver is buggy or misconfigured, and it isn't waking when the OS tells it to, because it is "mistreating" the screen wake event from the API as a synonym for the wake command (due to incompetence, poor design, laziness, or maybe just a bug)? Or, perhaps most likely, you don't have the magic packets coming through your network to achieve WOL?
This is typically a non-trivial task.
Servers
like to be on all the time. That's why data centers gobble down power like MRI machines. I think you can do it, or you can with Windows at least, but it takes some elbow grease.
BUT... That's a valuable test because maybe something
completely unrelated to the sleep/wake state of your machine is happening? Perhaps the mouse wiggling fix is incidental, or required for a different reason...
I'd test (and report the results of individually):
1. The default, built-in audio device on the Mini.
2. Sleep disabled.
3. Trying a
SMC reset (the universal "something is weird with my Mac's power or heat stuff" fix-it automatic doctor recalibration thingy).