Mini-Games:
Destroy the mentioned number of gifts. Afterwards, climb to the upper level of Mount Crumpit to access the chosen mini-game:

Spin 'n' Win - 750 gifts
Pankamania - 1500 gifts
Copter Race Contest - 2500 gifts
Bike Race:
Destroy 3,000 gifts and complete the whole game. Look behind the bookcase in the workshop with the sleigh to access the Bike Race mini-game.