Yes, it is possible to install and run DNN in a subdirectory.

I believe that the rights you listed above will be minimally sufficient to run an installation, but without "create" permissions, you may have some admin/host issues here in there.  For example, DNN would require create permissions to upload a file via the filemanager, or create a new portalalias.  It is possible to manually circumvent these issues (upload files via FTP, create the portal directories manually), but will require some additional skill on the part of the administrator.

Bottom line is that you will likely encounter some errors and headaches, and that these should be correctable, but will require some debugging skill on your part.  I haven't used the DNN installer before, so my advice here is more targetted to a manual installation.

Good luck!