In English: The double possessive construction [... is] extremely helpful, for instance, in distinguishing between "a picture of my father" (in which we see the old man) and "a picture of my father's" (which he owns). http://grammar.ccc.commnet.edu/grammar/possessives.htm
I don't think English needs the double possessive here; either "of the" or "boy's" will serve. And certainly there's only one possessive in the French.
The most literal translation for du/des is "of the". So "le chapeau du garçon" meaning "the hat of the boy" is the literal case. Sometimes we translate this as some, e.g. "je suis du riz" means "I have of the rice" where you imagine "the rice" refers to all the rice in existence, but in English this is better rendered as "I have some of the rice" or just "I have some rice".
Correct, "J'ai du riz." The main difference of am and have is that am refers that you are something as in a permanent characteristic of you and have is a possession or a transient being meaning you only have something for a period of time. You aren't rice so it would be "J'ai du riz" instead of "Je suis riz."
the French concept of gender is very different from English, very unnecessarily complex! even non living objects can be considered as female without any particular logic, so the adjectives get an extra e at the end to make them pointlessly look feminine even though it will never be pronounced which adds to even more irrational complexity!