OpenGL 2.0 isn't particularly modern (released in 2004), but its featureset is a good baseline if you want to make sure your application works on just about any even remotely modern hardware.
I suppose the "modern" in the title refers to the fact that the tutorial introduces vertex buffers and shaders right from the beginning, rather than delving into antique cruft such as immediate mode and fixed-function pipeline. This is indeed a good thing.
GLUT is pretty horrible (Does not support key bindings such as "Shift+Tab", does not report key presses when a different language layout is selected). GLFW seems much better, and simpler.
No, that version is not usable because the underlying API's (WGL and GLX) have since updated and using that ancient version of GLUT will not allow you to use OpenGL 3+ at all. There's also a bunch of other features introduced by WGL and GLX API changes.
The original GLUT was written for an OpenGL book (IIRC, the red book) so that the code examples wouldn't occupy so much space on paper. It wasn't probably intended to be used for real applications at all.
It still worked the last time I tried it a couple of years ago. If it gives you any trouble there is FreeGlut that is an OSS API compatible work a like. Its last release was this month.
I suppose the "modern" in the title refers to the fact that the tutorial introduces vertex buffers and shaders right from the beginning, rather than delving into antique cruft such as immediate mode and fixed-function pipeline. This is indeed a good thing.