Battle Damage is the damage inflicted to a player's Life Points from battle. Monster cards can still be destroyed even if Battle Damage is not inflicted. Damage from direct attacks count as Battle Damage.
0 battle damage, such as an attack from "Mucus Yolk" with 0 ATK does not cause damage, so no effect can activate in response, including those of "Mucus Yolk" itself and "Gorz the Emissary of Darkness"