-1) Freq. response - it will tell me if it's bright or bass-head, also spikes around 5-6KHz will make me mad alot.
-2) Square and impulse response for bass (anything between 50...100Hz should do) - will give me a good idea how well it will reproduce the low-end freqs. (usually round corners means diaphragm will not be perfectly accurate for low-bass), also if bass is "fast" or "slow" (phase delay is high or not).
-3) Phase linearity and impedance across audible range - will tell me if it will pair well with "difficult" headamps (high impedance amps usually, but tubes amps too or badly designed amps that might oscillate if impedance is not constant).
-4) Fit and comfort - this is very important and without a good fit all other measurements will not matter, especially on IEMs.
if I may...
2: SBAF has been making attempts to capture tone bursts at various frequencies which give interesting results but may be hard to interpret correctly.
Maybe even more interesting than squarewaves ?
Measuring squarewaves through resonators (ear canal) may be hard to do as not only FR is affected but decay as well.
I assume mad economist will know if that would be easy to do.
3: The problem with showing the impedance plot is that the only way this is meaningful is knowing if it varies and how much. You would have to get out the old calculator to get an idea how much the tonal balance would be affected.
For this reason I measure the actual response between 2 output resistances to get a feel.
A suggestion for Amir would be to use the impedance plot and then use different color traces, overlayed in one plot for say output R's of 0, 5, 10, 20, 30, 60 and 120 Ohm (perhaps even higher) to show real world effects.
4: I agree these are, aside from sound, very important and underevaluated aspects.
Fit and comfort are really hard to determine. That what person A finds comfortable may be horrific or not fitting for others. Take the wing design of AT. Most folks love it. I can't get it to work on my head. There are people with big and small heads, big and small pinnae, protruding or flat pinnae. For some people pinnae touching the drivers is hell, others don't care. Some hate higher clamping force others need it. Clamping force differs with headsizes, Headshapes differ. Boney vs almost round. Some prefer velours, others find it prickly and annoying. Some prefer leather or pleather.
There is little to determine objectively and when you do this would be very time consuming and subjective as well.
I suspect in order to churn out a lot of headphone measurements the generation of data and reporting would have to be done quickly and automated. The comfort part will be hard to do.