In the novels, what are the duties specifically of the lady of a castle – like what would Catelyn have done during peacetime at Winterfell? And where do these duties differ from those of the queen? Does a lady have more practical responsibilities, whereas the queen (it seems) merely entertains the court? Thanks in advance, love the blog!

The duties of a lady are to manage the household, to oversee and manage the estate (when her lord is absent), to oversee the education of the children, and so forth. 

The only difference between the lady of a manor and the queen is that the queen has the additional responsibilities of organizing the royal court and all of its social functions, but the queen also has more staff to aid her than the lady of a manor.