1. in terms of sound deterioration, how ‘bad’ is it to use a voltage step down transformer in general?
Not all step down transformer created equal, A good one will not degrade sound
quality, in some cases, an isolated step down transformer can improve SQ.
2. I wonder whether the Esoteric K05X can be modified to run on 220V? If so, how?
Depends on the model, some model just simply move connectors, some need to replace transformer, ask your dealer for advise.
I looked into the unit and I could see the ring transformer and on the main board some writing showing 250V.
Can you provide a link to some close up photos?