I think you can only use the unlockables in classic mode, which kinda sucks, but to use it you go onto the drop down menu where save load etc is and at the bottom it should say store, go into that and in your wallet will be any of the things you bought and you can add them into your game from there.
Its the only mode i know of that it works in, and just a warning, the son it generates sucks, might have just been mine but he sucks.
I dont know about needing the internet tho, but id imagine you would need the internet to use the unlockables but after that they should be like saved into your game, whether its online or offline