Linked just means they actively federate, allowed is a remnant from when the federation was strictly allow-list based in the early days (might be still possible to enable as a site admin), and blocked means exactly that.
The default is that any instance can federate, but only after a user actively requested that it is added to the list of linked instances. There is no central relay or so that tells an instance which other instances exists, so it needs to build up a list of known linked instances.
Regarding "federation state": You can have a blocklist, then Lemmy will federate with every instance except blocked ones. Or have an allowlist, then it will only federate with the allowed ones. If neither is set then it also federate with everything. Using allowlist and blocklist together doesnt make much sense, then it would only federate with allowed instances which are not in the blocklist.