Interesting project. Every office should have one.
The way I would approach this problem is to get the Arduino side working and debugged before you start on the GUI.
You're going to be using the usb serial port to control it from Processing, so why not design the data transfer format to be human-readable commands so you can control it from the Serial Monitor… then when it's working, processing can send those same commands from the GUI to the Arduino.
There are simpler approaches, but suppose you could send these commands to the Arduino via the Serial Monitor or a Processing program:
There is plenty of information here on the forum how to parse serial input and call your C code to work the motor and solenoid.
Good luck with your project,
Edit: grammatical correction.