Sure. It's going to be a pain, though.
If you consider DirectX or OpenGL as libraries, it's going to work only on your computer, since you'll have to work directly with the video hardware.
Sure, you could make a text-based game. People here will say it's not a good idea because it requires a lot of knowledge about parsing user input and string manipulation, but you could also learn a lot about game design by making one. I would suggest a library, but if you don't want to use one, try a text-based game.
Agreed. Not using GL/DX at this point is pointless. Computer hardware has become just too complicated and heterogeneous.
how would i make it compatible with other computers beside my own.
You would have to create your own version of DirectX, basically. If you're asking these kinds of questions it means you won't be able to do anything remotely like it for years.