Is it possible?
Almost anything's possible.
I want to get some inputs via the arduino from a melodeon with the reeds taken out. The digital inputs will be from the buttons on the melodeon and a pressure/vacuum switch (to sense bellows direction). The analogue input will measure the pressure in the bellows, maybe through a strain gauge.
This part is, if you wire stuff correctly.
The signals will go to the phone so that software on the phone can produce tones equal to the notes depending on button pressed and bellows direction in/out (+ve pressure/-ve pressure)
This depends on which Arduino you use, how you connect it to the phone, and how you program the phone. We can't tell how you intend to do that, so we can't tell if it is possible.
The user will listen via headphones from the phone so as not to disturb neighbors.
This might not be possible.