Why do you use "for" instead of "on" when you are talking about responsibilities? For example, if you wrote "they have these responsibilities for the house" why is it "for" not "on"?

Study these definitions carefully.

http://dictionary.reference.com/browse/for?s=t

http://dictionary.reference.com/browse/on?s=t

The best way of learning which prepositions to use is follow the language customs. Preposition definitions are often difficult to understand and apply.