Rewatching Ep. 31, I think I may have found Moofy's real name (Well, the name of who he was made from, anyway.) When Izumi sees the baby moving and then sees it's not human, she says "Aoya...!" Which, since it was left untranslated to anything, I assume is the baby's name. Either that or I'm way off base here and babbling nonsense.
(The sub spelled it Aoya, but it's pronounced more like "Aoi-ya". If that makes sense.)