When I mentioned cumulative distortion, I'm referring to how the energy present at a specific frequency in a multi-band signal is comprised not just the energy in the signal itself, but of it summed with all of the distortion products of lower tones. For example, the amplitude at 2 KHz is not just the 2 KHz component in the signal, but also includes energy from the 2nd harmonic distortion of the 1 KHz component, the 3rd harmonic of the 666.66 Hz component, the 4th harmonic of the 500 Hz component, etc. That's what I meant by cumulative: the level at each frequency depends not just on what's in the signal, but on the distortion components of lower frequencies that are played at the same time which lie at the same frequency.

In theory, if you could exactly match the waveform seen at the eardrum, then yes, you would hear exactly the same soundstage and imaging. However, I have never been able to properly do this in practice with over-ear headphones. Additionally, I've heard from an acoustic engineer that soundstage is partially influenced by physical factors; if the headphone is touching your ears, it hurts the illusion of soundstage because your brain knows that the sound is coming from right outside of your ear. In general, the brain prioritizes non-auditory inputs. That's why the McGurk Effect exists: when there's a conflict between what your eyes see and your ears hear, you literally hear what your eyes see, even if the actual auditory input doesn't match.

Regarding overshoot, in theory, less overshoot means it's more accurate to the input signal. That's what Dan Clark says to justify the macrodynamic performance of his headphones; he says that other headphones overshoot in their impulse responses, but his headphones do not. Many people think his headphones sound really dead and lifeless as a result, but that's where science meets art. If the music was produced on gear that has more overshoot, it probably has lower dynamic swings in the signal. Should the headphone reproduce the signal as is, or should it try to reproduce the dynamics that were in the original performance, but weren't mastered into the signal? There's no single right answer to that question, it's a matter of design philosophy and preference.

Regarding minimum-phase, it means that the phase response is exactly the amount needed to produce the frequency response. There's no excess group delay across the entire frequency range. In theory, the CSD plot shows nothing that isn't already in the FR, and generally weirdness in the CSD plot is reflected in peaks and troughs in the FR graph too. I used to hold strongly to that view, but now I'm not as certain that CSD plots have no value. All physical devices have resonances, and at higher amplitudes, those resonances are the first to exhibit serious non-linearities. In general, if I see a long trail in a CSD plot, then I take it as a sign that I should be very careful about boosting that region in EQ. If I cut, that's fine, because the trail disappears with the cut, but if I boost, then that trail will become more significant, maybe enough to become audible at high volumes. There is one case, the Koss KSC75, where I actually hear something like ringing in the lower treble (it sounds like a bit of microphone feedback when certain notes play), which is probably a resonance, but it goes away when I EQ down the 5 KHz region.

On the more practical side, I can only presume that phase is an issue at the design level. Many companies have tech in their headphones to control phase effects. Phase effects tend to result in weird, narrow peaks and dips in the FR. Those features become difficult to EQ out, and if you look at Oratory1990's EQ presets, he often doesn't fix those really narrow irregularities because they stem from phase effects that will sound really strange if filled in via EQ.

For your last question, speed isn't a real metric in headphones as far as we can tell. Some headphones certainly sound like they attack/decay faster or slower than others, but there's no concrete metric that can be exclusively tied to that phenomenon. Here's an article by Brent Butterworth at SoundstageSolo! that showed a perceptively slower headphone actually responding faster to the input than a fast-sounding headphone. Given my experience with dynamic EQ allowing me to add macro-dynamic punch to a headphone by overdriving large transients, I suspect that separation is part of "micro-dynamics" where small transients are being overdriven by a headphone. But there is no scientifically backed measurement that explains the perception of attack/decay speed in headphones.