Richard will offer you a second quest after you turn in at least 1 goblin heart, but that is not related to 'A hunting we shall go'. You should be able to turn hearts in as you find them, though.
The white rabbit should lead you north and then turn to the west to a chest that spawns on the river bank as you approach. The rabbit can be a little hard to see, but should stop and wait for you.
A few people have had trouble finding all the zeppelin parts to complete the quest, but I don't recall any issues of having a part and not being able to turn it in.
A couple people reported not getting Stone's Ring after defeating him in the arena (after the first encounter in his headquarters and following him through a teleporter).
The boss outside in Stone's fortress is Xanlosch (and his pet Isabelle). He only summons two flying demons if you destroy the generator there controlling the anti-dragon field.
