I currently have the mystery gift on my cartridge but was thinking about starting a new game. Would I be able to use that mystery gift on my new game or would it disappear or not be usable since it's been used previously and the event is over.
You should check out Pokemon Bank.
For $5 a year, you get 100 boxes in the cloud so you never have to lose a pokemon again. You can easily transport all your pokes from game to game and save to save (with exception of held items), given you have your own 3DS with your credentials on it and a WiFi-connection.
Pokemon Bank is compatible with Generation 6 and onwards (supposedly).