So far I've only met 2 of those requirements. The one I am currently using doesn't play back what's being recorded to tape, which is annoying when I'm trying to use it as an effect. It sounds decent though, and surprisingly can keep a constant speed (after applying at least a gallon of oil).
This was a suitcase style reel to reel tape machine, with detachable speakers. So I was recording the piano when I decided to take the machine apart, and I started fiddling with it, as usual. Then I took the piano apart, and started figuring out how that all worked. Basically then I put the speakers inside the piano, put the piano back together, and played my recordings of the piano.
I used a stool to hold down the sustain pedal and I basically made a reverb out of my piano. I think this would be a great way to turn MIDI piano into a more realistic piano. My problem with MIDI piano has always been the fact that when a note is played, the other strings don't react, reducing the resonance that a piano is known for. By playing your MIDI piano through a real piano with the sustain pedal down, you can record those strings resonating, and it should sound more like a real piano, because it is one!
Next what I'm going to do is get some speakers, mount them on the inside, and have an auxiliary input so I can play music from my iPhone into the piano.