To unlock New Game +, beat the game and wait through the credits. At the end, the game will prompt you to save. Do so, and the next time you start the game the completed save file will have a gun data icon by it. New Game + allows you to replay the game with all of your old gun data, and gives you access to several new endings.
Once you obtain the Girl Wings, you can use them to avoid random battles. You cannot enter a random battle while hovering. So if you repeatedly use the Girl Wings as a means of travel instead of walking normally, you'll never be summoned to play a random level.
Whether Scarlet appears in the ending or not depends if you pick up the Virus while on earth the second time or not. You can find it in the room above the one where Zart and Tierney were working. You cannot return here after you have left, so decide what ending you want here.
Only available in New Game +