One reason for multiple windows is if you want to use wglChoosePixelFormatARB to select your pixel format, and the final pixel-format might differ from the one selected with wglChoosePixelFormat for the dummy context. A window can't change its pixel-format once set.
As for the original question, do you mean that an active context claims to be 3.1 through GL_MAJOR_VERSION/GL_MINOR_VERSION, but functions that are guaranteed to be in core 3.1 are not available?
If they are extension functions, check if they are available in the extensions string.
EDIT: If I understand correctly the problem is that, when calling glGetIntegerv(GL_MAJOR_VERSION) and glGetVersion() after each other, on the same context, they return conflicting information. Which seems strange to say the least.. can you make a minimal example and post the code?
As well as update your drivers, might be a bug.
My guess is that glGetIntegerv(GL_MAJOR_VERSION, ..) returns an error (check with glGetError()), and does not overwrite the integers in int version. Therefore some old values that happen to be 3 and 1 are still there and the real version is 1.2, for which GL_MAJOR_VERSION is not supported.