Hi and welcome to Arturia forums.
The DX7 preset 'hopping ratios' - I tried to recreate it yesterday, using the Mod window and the matrix, but couldn't get it to work.
There are more to it than the Matrix settings.
In example you mention the modulation envelopes with the images, but they are not used. The Operators envelopes are used in the preset.
Is it possible to build it from scratch, or to tweak a different DX7 sound to have the same arpeggiated switching between different coarse tunings?
Yes. All the presets is a result of the settings anyone can set in the synth.
The random modulation is done by choosing the Sample/ Hold waveform for the LFO's.
Are the 'data entry' sliders re-programmable or are they set in place? Couldn't find a way to edit them.
What do you mean?
What could'nt you edit?
Is it just the DX7 that can be programmed this way? Is this a common feature of synthesisers? If so, what is the technique called, so I can look up more about it?
Many synths have Modulation Matrixes, and LFO's or other things that can create some kind of random modulation.
This preset use random modulation of the pitch.
You can find many informations in the manual. Also about using the Matrix and the Data sliders.