Next to correctly mounting your drive, you also need to set the desired ownership and permissions. Drives that are automatically mounted during startup (i.e., included in the configuration file /etc/fstab) are by default owned by the administrator ("root"). Depending on default permissions of the system, other users will have read access only, or may have no access at all.
It is normal (and actually desired) that you do not see the drive 'as such' in Files. In linux, you can link any storage from within the home folder. Thus, from the user's perspective, all storage is neatly in his home. It is only the administrator that needs to be aware that, in reality, the storage is under /hdd (in your case).
Changing ownership
You can change the ownership of the mount point (/hdd) to give the entire mounted partition to a user. You can also assign subfolders of /hdd, that means folders on the mounted drive, to a user or to different users.
To change the ownership of the mount point, use a command like:
sudo chown <user>:<group> /hdd
Substitute the login name of the user and the group that needs to be owner. On Ubuntu, each user also belongs to a group of the same name, so may then be the same login.
Changing permissions
After changing the ownership, there will be no need to change permissions. Each folder has permissions ("read", "write" and "execute"). By default, owner and group have all permissions, "others" can read and execute, but not write.
Linking your folders
Indeed, you can move the data folder to the hard drive, and replace them with a symbolic link that points to the new location. From a user perspective and for practical purposes, such a link acts and feels like a real folder.
