Pretty certain that the second set of wires (black) is to a vacuum release solenoid - otherwise the unit would create a constant vacuum on the breast, rather than a suckling action. They are PWM'ing the solenoid as well.
I would also guess that there is no pressure sensor in the unit, or calibration required (they will have a vacuum test-jig as part of final test). The 24C01 may record cycle counts for service feedback, but it's main goal will probably be to remember the pressure settings since last time. Perhaps Dave can monitor the I2C packets with his MSO and see when the unit reads/writes to the memory.
The unit can be powered from an external mains adaptor, so the non-reset nature of the fuse is probably wise, also - it may a zener diode to protect against over-voltage (although I doubt it, I would design it withstand 15v and have a processor 'over-voltage' detection routine).
The black foam is probably a dust-filter for the vacuum release solenoid.