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 return to Kurogane. He will 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 accessible by defeating a boss in certain conditions
||How to Unlock
||Kill Chimarea when its poisioned
||Kill Harupia when she is poisioned
||Kill Hellion with death status ( Issen, Climax)
||Kill Colossus without bashing it ( No bash element attacks)
||kill Scylla under or in 5 turns