Status effects on either parent can get passed onto offspring, including things like poisoned, unconscious or confused. They can be cured the normal way and may only affect the first battle, but get added to the heritage page and passed on.
Blank heritage entries sometimes seem to get added to the list (may be related to previous bug if these are hidden states getting passed on).
Heritage perks don't seem to be reset when the game generates a new character. Recruiting a new character from the guild can generate one with the same perks as a character previously dismissed. Seems like this may also be causing those attributes to stack much faster than they should and well beyond the limits they're supposed to have (I've got characters with 63/10 in LUK+, for instance)
Defeated enemies can be fought over and over again if you use the action button after battle before they vanish from the world map.
This isn't quite a bug, but rather a flaw in the new player experience: it's not clear what the expected lifespan of party members should be. A simple fix could just be a town member mentioning that they usually expire at around 80.