Extra outfits:

Beat the game on normal or hard difficulty to get a Closet Key. Use this in the room where you found the Hunting Gun to get new costumes.

Leech Hunter mini-game:
Beat the game with a B, C or D rank.

Submachine Gun:
Beat the game between 3:31 and 5 hours for an A rank to get the Submachine Gun and Leech Hunter mini-game.

Rocket Launcher:
Beat the game in 3 and a half hours or faster for an S rank to get the Rocket Launcher, Submachine Gun and Leech Hunter mini-game.

Leech Hunter Rewards:
100 Leeches: Unlimited ammunition for any weapon.
90-99: Magnum Revolver in room 202 of the Train.
60-89: Unlimited ammo for the Hunting Gun.
30-59: Equipped handguns have unlimited ammo.
1-29: Submachine Gun bullets added to Rebecca's inventory.

Dail door number puzzle:
Enter "4863" on Disc 2, where the cable car is located.

Chess puzzle:
Arrange the big pieces in the same order as the pieces on the smaller chess board. The difference is the only white piece by itself. Use Billy to push it down, right, and finally right in front of the black piece. The desk will now open.

Brake system puzzle:
The main code is 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 7 + 2 = 81. This should activate the brakes for the train. The second code for the brakes is 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 5 + 6. This will start the brakes.

Fountain circle puzzle:
When you get to the fountain circle with the six animal statues on Disc 1, use Billy to light the statues in the following order: Deer, Wolf, Horse, Lion, Snake, Eagle.

Thanks to Revolution readers Carlos Aparicio, c0venant_l33t, and zac conrad!