About the 2D/3D: most 2D/3D related calculations can easily be extended (A 2D vector addition can also be done as a 3D addition with the last element zero) and most API's accept 2D data as well. This means that you can perfectly use the OpenGL/Direct3D or any abstraction (SDL/SFML/Allegro...) for 2D games and still utilize the power of the GPU. Given that a GPU has a raw processing power that is far beyond a CPU's, even for simple 2D games like platformers I'd recommend using it.
About the "do I need a videocard"; the whole pipeline goes a little like this:
Application <-> (optional) Abstraction layer (such as SDL) <-> OpenGL/DirectX <-> Video Drivers <-> Hardware
This means that if there is a video driver that uses the CPU instead of the GPU; then you can run a game without a videocard but given the fact that the CPU has a lot less processing power (but is a lot more flexible) it probably won't be very smooth.
The hardware (GPU's) are there because they offer enormous amounts of power,
The drivers are there because every hardware is different and every vendor has their own architectures, (and because of this writing code directly for one GPU architecture is a very bad idea!)
The OpenGL/Direct3D api's are there because they standardize calls: you don't want vendors to include their own special features so you have to write your graphics code only once to run on all GPU's that support a certain OpenGL/Direct3D level,
The abstraction layers are there because the OpenGL and Direct3D API's are suited for all possible graphics applications and libraries such as SDL make certain applications easier (and often provide other features such as input handling which a graphics API doesn't provide).
Edited by Miklas, 20 September 2013 - 02:24 PM.