I have used my AKG K702 headphones for a few years now and I absolutely love them. These are truly great headphones.
I will now comment on the issues you have raised.
Well I haven't heard the "magni/modi" that you have so I have no idea about that.
My personal preference is for solid state amplification. Some people might suggest valve amps (known as tube amps in the US) but my own experience is that good solid state amplifiers are more likely to provide the neutral and transparent input for the K 702s. My experience of valve amplifiers is that they invariably add colouration and you really have to spend a lot of money to escape that.
The K 702s do have one problem which can be exacerbated by some solid state amplifiers. The treble can get out of control and very overemphasised on some recordings. I find that with the K 702s the treble is really excellent as long as it does not lose control. However on some recordings it will lose control and then it is a problem. So when choosing solid state amplifiers avoid those that over emphasise treble. Some people suggest valve amps for the K 702s because they will often dampen excessive liveliness in these headphones, but I believe that the sonic costs with valve amplifiers are too great, so better to get an appropriate neutral and transparent solid state amplifier with well controlled treble.
Personally I suspect burn in is much more to do with the psychology of the user than the headphone. I would love to see some comprehensive tests on this alleged phenomenon.
I think the best thing to do with the headphones is to listen to music with them and enjoy them. If they do improve over time in some way, then well and good, but as they sound just excellent out of the box, then just use them and forget about burn in.
I use my AKG K702s with the cable that they came with. I would be very surprised if replacing the cable with some other one made any noticeable difference in the sound of the headphones. This is something which is argued about a lot at these forums, some people claim that these cables make a difference. Human hearing is extremely susceptible to auto-suggestion and I suspect that the differences being reported with different cables are just auto-suggestion.
When those people who claim there are differences demonstrate this in blind ABX tests then I would start to take notice.
I first started listening to music with Hi Fi in the 70s and I have been doing so consistently ever since that time. In the 80s I bought into the whole cable thing with Hi Fi, it was a new thing then. I wasted some money on cables then. It was only years later when I realised that those expensive cables made absolutely no difference to the sound of the my Hi Fi.
Well I listen to classical music mostly and if you do decide to check out some orchestral material you will be truly amazed at how good the K 702s are with this kind of thing. I love the impressionists mostly, people such as Debussy and Ravel. Impressionists really use orchestras very inventively to create new types of sound. These extremely intricate soundscapes with their inventive rhythms and many tiny details are reproduced by the K702s with superb accuracy. It is amazing how good they are given the price they are sold at.
I do listen to other music as well. World music, English folk, Jazz and good old rock/pop. So hard to recommend anything specific. The K 702s will produce any music very accurately. I love the bass in the K 702s, it is true that they will not do thick heavy bass, but they have such well defined bass that listening to any bass player, either with an electric bass or a double bass, it always a joy.