If I understand things correctly:
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?
I think you're confusing two versions of Hive, which are really completely independent and incompatible.
Hive Mac (which this thread is about) has (for now) no 12-word seed and has only one address, and to restore the funds later you need at least one backup, e.g. in Dropbox or Time Machine (technically it doesn't have to be recent, though with an old backup you'll lose some transaction metadata).
Hive Web/iOS has a 12-word seed phrase and auto-generates a new address automatically from the seed for every transaction. To back it up you only need the phrase and that is enough to restore the funds and all previously used addresses.