Do I only need my wallet seed (12 words) to recover my funds (and addresses; there is currently one address in the Hive wallet but I guess more will be added) from 3rd party sources, or do I need the actual address or addresses as well?
You only need the 12-word passphrase to recover your funds. You can learn more about passphrase, PIN and addresses here:
https://github.com/hivewallet/hive-js/wiki/Hive-Web-FAQ