The main purpose of this page is to discuss conceptual questions related to the course material, in order to enrich your understanding. Increasing the charge will proportionately increase the potential difference as well.
Capacitance depends only on the geometry of the capacitor.
During lecture I did a demonstration using a capacitor with a fixed amount of charge. I showed that if we increase the plate separation, more energy will be stored in the capacitor. Since the plates are oppositely charged, they attract each other. That energy goes into the capacitor. Note that although the charge and electric field stay constant, the voltage between the plates increases as we increase the plate separation.