I think the problem with the invalid links can you solve in this way:
1. Create in the Backend a new menu and name it 'Hidden' (see the first three pics below)
2. Create a new jDownloads menu item and select as menu module the previously created (fourth pic)
As result should be now the links in the tree module (and some others) correct.
Please note that normally must always exist a frontend menu item for jDownloads. But it is not a must that he is visible.
Please tell me the result and also what happens with the missing categories.