So it's been a bit less than 6 months since I last updated this thread, and I thought it time to contribute a bit more on this subject.
I've done several things to try to iron out the issues:
1. Added a linear power supply to the streamer (Small Green Computer LPS)
2. Changed the ethernet cable to CAT6a
3. Added a Stack Audio Smooth Lan just before the streamer
Things are perhaps slightly better, but really still not up to the standard I'd expect from all of this. Still having the same issues. I'm beginning to think that the problem may indeed be with the streamer itself (it's considered entry-level, after all), but probably not with my DAC (to remind you, it's a Sonnet Morpheus MK1, connected via USB to the streamer) because, as mentioned in one of my previous posts in this thread, CDs, DVDs, and anything that comes through the Roku box sound very good indeed. While there is an option to connect via I2S, my DAC only does this via ethernet cable, and that's not an option for the Innuos PulseMini, which can only do this via HDMI.