well if you dont want features to show on cats and photos pages then under show photos options in admin the option to show features on the page set to NO.
As far as the categories like previously stated once you make those cats top level you then define subcats for your parent cats. Then you can hit the parent cat and get a list of subcats
I will show an example
My gallery it setup more like what you had but lets try and follow this A community photo gallery - ReefTalk Gallery
Obviously I list subcats depth to 1 but anyway lets pretend it was 0 and the only thing that showed was reeftalk gallery and members gallery. If I click reeftalk gallery I get this ReefTalk Gallery - ReefTalk Gallery
That is the exact nature of what you saw on the other site click the parent and get the subcats list.