This can't be done with the Moodle custom menu the way it is coded at present, although I suspect it could be made to work like you want it without having to 'clone' the theme.
Basically, to make the custom menu work, the way you want it to, first it will need some id selectors adding to the various parts of the menu structure. And then some code putting in the renderer.php to govern the way it works. Easier said than done.
Making clones wont solve anything, as the custom menu is site wide, so what's in the menu will be written into the d/down menu on whatever page your site dictates it should be available in.
You could create a new layout page for course/category (you only need to make one) but it would have to be declared in the Arctic's config.php file.
This layout could have a separate (manual) dropdown menu which could be included into the course/category page. This would follow the same structure of the YUI dropdown menu in Base YUI (Moodle 1.9 Theme) by Patrick Malley.
Depending on what you want to add to each course/category page would need to be determined as it could be quite a time consuming thing to create.
What Richard suggest might be a better option. As all you would need to do is add the basic structure of the menu a an unordered list and then add some style to it in the css file for the theme.
You might like to look at the type of menus you could use HERE.
Hope this helps?
Mary