The blocking of "root" access to FTP is most likely a security restriction at the server end in the FTP deamon (ftpd). It is almost certainly not a restriction in your FileZilla FTP-client application. It is a bad idea to allow root FTP access because FTP passwords are passed over the network in plain text and are not encrypted - curious strangers can easily get root access to your server. (you should probably be using ssh and scp or sftp instead)
The reason your new user cannot delete files from other directories is that the new user does not have write permissions on those directories. If you want a stranger sniffing your network to be able to delete everything in a folder just give new user write permission on that directory, typically by changing the group of the directory to one that new user belongs to (chgrp groupname directoryname) then setting permissions (chmod g+w directoryname) Note that this may cause problems with other applications and is inadvisable for any directories outside user's home directories unless you really know what you are doing.
So far as I know sudo and visudo have no effect on permissions when using FTP.