Great questions.
Royal granary would rot due having to travel to the capital, plus im sure the sewer rats of the cities would eat it rather quickly.
The idea for access for the dukes is questionable yes. However, the Ruler determines who gets access and the amount of access. If Ruler trust the duke then yes, you'd assume they'd have some access. Most likely regulated by the Ruler or perhaps Banker since his job is to handle finances for the Ruler as well as the realm.
