At the opening scene of Xerneas/Yveltal, press the following button combination: Up + B + X. This should prompt you for deleting your save file. The Up + B + Select no longer works on the X/Y series at the main menu, the button combination must be done at the opening movie sequence.
The legendary bird you can catch is based on which Pokemon you picked as your starter. Choose Fennekin to get Zapdos, Froakie for Moltres, and Chespin for Articuno. There is no way to find all the legendary birds at once. If you choose Fennekin to get Zapdos, and then you transfer Zapdos and delete your save file to choose a different starter, then eventually you'll have all the birds. You have to beat the Kalos League to even make Zapdos/Moltres/Articuno appear. You'd be better off using the Pokémon Red/Blue/Yellow Virtual Console to fill your Dex rather than playing Pokémon X/Y 3 times. Once you encounter Zapdos/Moltres/Articuno once, it will instantly flee, not giving you a chance to fight it. Zapdos/Moltres/Articuno will appear on random routes after the player beats the Pokémon League. It will continue to flee for a total of 10 times, after that it will be at the Sea Spirit's Den, where you can battle and capture it. It will be level 70.