Also, is it correct that unlike DirectX, it is not possible for a game to install OpenGL, so OpenGL can only be installed with the drivers?
Sidequestion: are any of the big games still OpenGL these days, or is Rage the only AAA OpenGL left?
That goes for part of DX aswell, DirectX is basically 2 parts, the DirectX runtime and the driver, (Microsoft provides the runtime, the IHV provides the driver, allthough Microsoft does provide drivers for most hardware through Windows Update aswell). (IIRC Microsoft does some quality assurance on D3D drivers aswell to ensure basic compatibility)
For OpenGL the IHV provides both the driver and the runtime allthough Microsoft does provide a basic OpenGL 1.1 software driver and runtime to get basic support no matter what.
One likely candidate for the Win7/8 problems could be that the machines are using multiple GPUs, alot of newer laptops have both an integrated Intel GPU and a nvidia GPU, it is not impossible that the integrated GPU lacks OpenGL drivers while the nvidia one has them and that by default your application tries to use the Intel GPU. (installing drivers manually from the IHV rather than from the system builder might override this behaviour)