The Wii U can store download versions of games, but there are only 32GB available, once that runs out you would need to use an external hard drive (and Nintendo recommends using one with a separate power source.)
The GamePad screen's use depends on the game. Some games allow a single player to use it without needing a TV at all, some games may require using both the screen and a TV. In Mario Kart 8, I believe the only options are to mirror whatever is being shown on the main screen (including split-screen multiplayer), a button to honk the horn, or an overhead map of the course. Someone can correct me if I'm wrong. 