Is the piano new enough to have a USB port?
No. If it was, it'd still be supported

Basic USB/MIDI converters (sufficient for this job) are about $8-10.
Does this thing have fancy weighted/levered keys that are designed to feel like real piano keys?
Yes, it does.
Does the noise appear to get quieter or go away if you turn down the volume?
Yes. The piano also has a key sensitivity feature. If I set it to "firm", I seem to have to hit the key harder to trigger the distortion. That suggests to me that it's independent of how hard the key is pressed, and -- potentially -- of the key switch entirely.
Oh, here's an idea: the piano can record its own sequences. Could just use that. No need for MIDI. Doh.