You cant walk into class's, actually other than the homework tasks (in which you have to find the right book in the library then do the 'exam' for your O.W.L's) there is very little to do with the class's, most of the spells you learn are when you are 'teaching' them to the members of DA. I have not come into contact with brooms or quidditch as of yet, but I am only at the 46% mark.
Yes you can explore into the forest and the Owlery, Hagrids hut, etc.
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