Book Of Lost Memories:
Complete the game successfully, then start a new game and took for the newspaper near the Texxon Gas Station. You'll find the "Book Of Lost Memories" inside the newspaper.

Dog Key:
Complete the game with the rebirth ending successfully, then start a new game. Near Jack's Inn you'll find a dog house in wich you can find the Dog Key.

Obsidian Goblet:
Complete the game successfully, then start a new game and go into the Historical Society building where you can find the Obsidian Goblet a shelf.

Extra options:
Press L1+R1 at the options screen.

Disable scratchy filter:
Complete the game successfully and save it. Press L1+R1 at the options screen.

Extra ammunition:
Complete the game successfully and save it. Press L1+R1 at the options screen.

Another hard riddle mode:
Complete the easy, normal, and hard riddle mode in the game successfully.