To finish the Cerberus Cup, I suggest equippin one of your drive forms with the Wishing Lamp. Also equip Jackpot on Sora and Draw, as well as a draw ring. When you defeat an opponent, more orbs will fall and draw allows you to absorb them quickly.
To unlock some recipes, apply serenity materials to ones you have already synthesized. For example, applying a serenity crystal to Save the Queen unlocks Save the Queen+.
New recipes are also unlocked once the moogle reaches a certain level. So if you're moogles already level 9, all the recipes you haven't unlocked, will be unlocked by synthesizing recipes with Serenity materials.
Also, I'm not quite sure about this, but more recipes are unlocked when you complete a certain synthesis goal (collect 100 of this, 100 of that etc.)