After I deleted an application from my iPhone, is there a way to find how many times that application was deleted and reinstalled on my account?
Shot answer is no. Long answer is why?
There is not a method to see that. Unless you can go on your router and locate the logs from about when you got it from. That would be a very hard process, nearly impossible.
In your iTunes account you can see Purchase History, including free apps. It won't show when it was deleted, and may not show re-installed apps.