several relevant points:
1) Changing the category  in "Select a portal category --" does not 'move'
the document  library. It simply produces and 'listing' reference in the
other Areas you check (and you can uncheck the 'Home' location). The
'listing' is just the url and it can by displayed on the new Area by adding
the "Grouped Listings" web part.

2) It is always a good idea to give every "Document Library" you create a
unique name (easier to manage:
-->Site Settings, -->Manage security and additional settings, -->Manage
storage space allocation)

3) If you subsequently delete the 'Document Library' the 'Listing'
references remain in the locations where you selected/added portal
categories.

4) You can control 'permissions' at the 'Document Library' (adding/selecting
Interest Groups) and/or at the Area/Subarea using 'Manage Security' (-->Site
Settings, -->Manage portal site  structure, 'Manage Security is part of the
edit drop-down on the Area listings).

5) The best place to control 'permissions' to all navigational elements on
the portal is -->Site Settings, -->Manage portal site structure, -->Under
the "Views" click "All Areas".