According to my computer's dictionary, the official definition of hermaphrodite is: "First appeared 14th Century; 1: an animal or plant having both male and female reproductive organs. 2: something that is a combination of diverse elements." Given that definition, I think that some of your questions have been answered. At birth many parents determine what sex their hermaphrodite child will be--though some cases reverse themselves later in life as gender isn't only about sexual organs.
I hope this helps.