Extra Ring Slot

[Togainu]

To get the third slot "working". Equip the sword and go to the in-play tab to enable the third slot (basically indicating you spend the time to put a ring in etc). At that point you can equip a third ring.

As for having to equip it. The sword talks about having to wield it as stated here.
Once a ring is fitted to the tenth slot, the sword’s wielder can employ that ring as if he were wearing it, even if he is currently wearing two magic rings
Meaning just carrying it won't do anything seeing you aren't wielding it at that time.