Remember "melt starts" at the Spring zero crossing and ends at the Autumn crossing, so no, the charts do not suggest a change in timing of melt. Likewise, drops in both maximums and minimums would (all else being equal - which they are not) mean lower daily change rate magnitudes rather than higher. It is tough to eyeball the effect of the rate chart on the amount of sea ice.
The Spring zero crossing is chaotic, the Autumn (weather filtered out) is strikingly consistent. There has been a lot of discussion the last few years over whether peaks are earlier or later than the past. Provisionally, I believe neither has changed markedly. The excitement comes because a weather change can readily skew the peak by a week or two - given the fundamental rate of change at maximum and minimum is small compared to the rate of change a large weather blip can create. A few of these in a row can look like a trend, even when they are not.
The charts below are from the full NSIDC record, filtered to essentially exclude changes with a "period" of less than 15 days while creating minimal distortion (visible in the last graphs I posted earlier, if you look for it - I've modified the filter) in changes with periods greater than 30 days. The first is an enlargement of a random chunk of the record, to clarify the annual detail. The second shows the entire record.
I can generalize that every year, for many years, the daily rate moves from peak melting to peak freezing in a smooth fashion at a brisk and increasing rate. The fall in rate is slower and much more chaotic - at this level of smoothing all you can say about the shape is that it takes much longer than the rise in rate did. If you look closely, this pattern is visible every single year, always chaotic from peak freeze to peak melt and with at most a glitch or two through the period in "the groove" from peak melt to peak freeze.
As an aside, the plots of summary charts seemed to indicate that, over the years, the rate settles down onto "the groove" sooner and stays with it longer before becoming unstable. I caution this is wild conjecture until I look a lot harder at individual years.
This is premature, but I'll throw out two of the factors I believe are involved. The first is geography. The smooth part of the curve roughly correlates with the period when the action is happening in the Arctic Ocean proper, away from land (other than the CAA, which in the old days never melted anyway). The next rough correlation should be near and dear to Rob's heart - and he knows a lot more about it than I do. I believe it is fair to say that in the smooth part of the curve, most of the sun that falls on the annulus of active extent decrease is falling on water - not snow, land, or ice, but water. In the chaotic portion the opposite is true.
The sunlight to the atmosphere is identical on the two equinoxes. I don't know enough about cloud climate to know about the sun hitting the ground, but speculate it is not radically different either. In the Spring however most of the sun falls on ice or snow, minimizing its contribution to the heat balance. In the Autumn, most sun falls on water - its contribution dominates the heat balance. In short, the chaotic portion of the annual trend is controlled by heat arriving with ocean currents and air/humidity in weather systems, which are of course as fickle as the weather. The smooth portion is dominated by insolation, which is (astronomically) dead consistent. The mismatch between the peaks in the curve and the solar equinox calendar is driven by geography and melting - until the sun is consistently heating water, it can't drive the problem (wresting the reins from weather).
I have not seen a formal heat balance of the Arctic (even on crude terms) but suspect it will eventually have a lot to say about the shape and timing of the daily rate curve.
FWIW, global warming affects both the ocean/atmospheric and solar heat inputs directly. And again, I as I conjectured, it may affect the timing (changing the date at which snow, land, and melt ponds take their part in the process). The effect of the latter on SIE overall may be one of those non-linearities we've been looking for WRT CO2 forcing.
I don't think we will ever be able to predict how deep next year's jet stream waves will be or where on the ice map they fall. I fear we will increasingly be able to predict changes in melting within the Arctic basin proper (which will become more and more of the ice year).
So, Rob, I am intensely interest in where you have been going. I suspect that if enough data exists to put a statistically effective "years through time" factor (regressing wrt year rather than averaging across years) into each term of your equation from July of 2013, you may find something that works more rather than less well as years progress. It is also possible this "snow/extent/area" approach may yield an effective prediction sooner and sooner in the year.
One thing I've been scratching my head over is "extent". I can see consistency in ice volume rate (linearly related to heat movement), but why we see such consistency in extent rate (at extent minimum), given the enormous (annual and trending) changes in the size of geographic area where the extent change occurs (roughly proportional to perimeter length of the MIZ) and thickness of ice at and approaching the edge has me buffaloed. Robust statistical trends are lovely, but in physical systems they have to have physical explanations . . .