BGM Option
To unlock the BGM option you have beat the game. The BGM option can only be used in-game.
Unlock Beast Class
After you defeat the 5th floor boss go back to where you met Kurogane and he willl give you the Trust Band. Go to the Explorer Guild and you will get the Beast class.
Unlock special items
These are the boss items that are only acessible by defeating a boss in certain conditions:
Brute wing - Kill Chimarea when its poisioned.
Cut Quill - Kill Harupia when she is poisioned.
Red claws - Kill Hellion with death status (Issen, Climax).
Red Spine - Kill Colossus without bashing it (No bash element attacks).
Snow bone - kill Scylla under or in 5 turns.