1. I found that the sound does significantly improve when plugging them using the provided cable vs. over Bluetooth (could be that I get to use my dedicated amp/dac v.s. whatever's built into the headphone?). I do think you lose the ability to have ANC with them plugged in though, so this option is best used at home.
2. I have the Beyerdynamic DT770 Pro, the Sennheiser 6xx, and the Momentum 4. Compared to the DT770, the Momentum 4 has less punch in the bass, and slightly less emphasized highs, but the M4 has warmer mids and sound overall more detailed. Compared to the 6xx, the Momentum 4 have more bass and more highs, but are less detailed, especially in the mids. Overall, I'd say, DT770 <= Momentum 4 << 6xx, but that's just my preference. I think the DT770 and Momentum 4 are pretty even when plugged in, just depends if you want more of a V shaped sound with punchier bass (DT770) or a warmer sound (Momentum 4). 6xx has detail that just far exceeds both of the others.
3. I did experience software bugs when I first got my M4. They failed to pair, would pair and then not play any sound, and there was even a time where they wouldn't turn on or respond to any button presses at all. I was able to factory reset them to fix the issues, and since the most recent firmware update they have been acting as they should with no issues. Definitely update them as soon as you get them though.

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Overall, I like the M4 when I'm on the go as they have ANC and Bluetooth. I rarely plug them in, but I think they'd make a great can if they are your only higher end headphone and you need to be on the go.