Frequency Response as flat as possible? Nope, this is why I dislike measurements in the first place, the FR of a headphone is affected by something called HRTF, head related transfer function, which describes how sound reflects off the geometry of the head and ear. Everyone's head, ear, and inner ear geometry is different, and because of this, everyone hears things differently. "Flat line" is not a flat line for everyone. There's no way to account for this in measurements, you just need to listen to a headphone you're interested in and decide for yourself.
Harmonic Distortion doesn't matter? Are you stuffting me? THD is very important, if a region has high THD, it will sound thick and heavy, how does that "bear little importance to overall sound quality"?
Isolation, sure, close enough.
Impedance, while it's good to have a flat impedance, the phase is what really matters, and tbh having higher z is usually a good thing, as the output z of the amplifier you're using can interfere with the bass response if the headphone's z is too low. It's called damping, ideally, you want the output z of the amplifier to be 8x less than the headphone's z. A lot of amplifers are poorly designed or have lax design, leading to output zs of greater than 1 ohm, tube amps especially, though tube amps in general are a debate for another day and another thread. When the factor of 8 is exceeded, the headphone's diaphragm starts to lose control in the bass, and as the output z of the amp increases, the effect affects higher and higher frequencies. I would go into reading the impedance, phase and sensitivity for the sake of choosing an amp that will power the headphones correctly but that's a 10 pager that doesn't belong here.
Square Wave, nothing will ever be square, and in fact, having a moderately strong leading edge and fast recovery is better than looking like a square. As tak said, after that it should be a smooth line.
I would like to say I put no water in measurements however, as I've experienced, headphone audio in particular is too subjective to be portrayed as graphs and statistics. I can't get any sort of feel for the differences I hear between headphones with graphs; what I see is not what I'm hearing. There's a fundamental flaw in measurements too: headphone measurement is a very new thing, there are more than 4 different schools of thought as to how headphone measurements should be taken, what sort of corrections to apply, how to interpret them, etc, and nobody can decide on or agree on anything. This is a red flag for me, and a big reason why I advocate listening instead of trying to get something meaningful out of graphs. Also, those graphs look to be Headroom graphs, which were taken on Tyll's old equipment using worse methods, it could stand to be redone with the graphs he's made for InnerFidelity.
