You will need items in the category for it to show up in your store listing of categories. To move active items into categories you will need to revise the item and select the Store Category for the item in the listing. You can do this through Auctiva > Listings > Active. Just select the item you wish to revise and then click the Revise button at the top.
This does not answer all your questions, but this is the 1 I have time currently to deal with. (It was the simplest!)