Hard Mode:
Beat the game on Normal difficulty and save after the credits. Now at the Title screen, hold L and start a new game.

Start with all elves:
Beat Hard mode with an A or S ranking and save. Now start a new game using this saved game to begin with all elves.

Get the Z Saber:
After Ciel gets caught by the Golem, go follow her to start a boss battle. After about a minute, Zero will curse and you'll see the the computer activate and give you the Z Saber.

Get the Triple Rod:
After you complete the "Retrieve Data" mission, go to the basement inside the Resistance base and enter the first door you see. Talk to the person inside to get the Triple Rod.

Get the Shield Boomerang:
Once you've beaten (or aborted) all missions except for the last two, go to the basement of the Resistance base and talk to the same guy who gave you the Triple Rod. He'll now give you the Shield Boomerang.

Thanks to Revolution readers Dylan Kuehl and Barson Barson!