I have flat feet, too, but haven't experienced what you are experiencing. I do get a tightness and pain in my feet when I wear certain shoes that aren't shaped right for my foot. I realized it was because the shoe was not allowing for proper movement of my foot. The best sneakers for me are the Nike Free running and training shoes. The worst have been Nike Air Max running shoes, I think because the movement is too stiff. It also happens with new dress shoes, but goes away when they break in.
As far as your calf exercises and getting cramps in your feet, that sounds like it may be more of a form issue than a flat-foot issue. I could be wrong as I'm not an expert. Is it possible your movement is too focused on the foot doing the work versus your calf muscle? At the top of the movement, is your foot flexed, or is your calf muscle flexed? I always keep my foot pretty relaxed during the exercise and really focus on my calf muscle and never have a problem. Also, for my flat feet, I've found that pointing my toes outward during my stand-up calf exercises really emphasizes the inner gastrocnemius.