What are the libraries I need to achive goals 1 + 2? And can someone point me to a tutorial where loading and using shaders is explained well. Please no NeHe, I am getting confused by those tutorials.
You do not need any of those libraries. Nonetheless, they may make your life easier.
GLUT, FreeGLUT, and GLFW are libraries that abstract away the part where OpenGL touches the operation system. This part is not a component of OpenGL itself. If you deal with pixel formats, window handles, and input events directly in terms of your OS of choice, then you do not need them. Such libraries are neat in case that you want to go cross platform. IMO from those three you should go with GLFW (AFAIK GLUT is obsolete anyway).
Another library is GLEW that supports you in working with OpenGL extensions. This library is not necessary for the first steps (EDIT: Seems that for Windows this library is usefulin any case /EDIT). It is mostly meaningful in case that you want to write programs that should run on many platforms where some functionality may or may not be available from extensions instead of the core of OpenGL.
Another library is glm with provides vector and matrix math for OpenGL. It is obviously useful because with the go away of the matrix stack you need to do the math yourself.
What about Vulkan? Is it worth learning OpenGL right now? When is Vulkan coming out? Will it differ much from OpenGL? I read that it will be even more low level than OpenGL? Or would it be smarter to wait for it instead of deep diving into OpenGL right now?
There is an evolution from OpenGL 2.x over 3.x and 4.x up to Vulkan: Things got more and more compacted and closer to how the hardware works. People who know this evolution will probably understand Vulkan quickly. But I assume that jumping from immediate mode to Vulkan is a bit difficult. I would suggest that you start with OpenGL 4.x especially if you want to start just now.
If I get it right OpenGL applications can be implemented in both C and C++, what should be my choice here? I have a feeling that most of the tutorials are in C++? What are the pros and cons here?
OpenGL for C/C++ is a C API, so both are possible. You can also use Java or any of the various other language bindings that exist. However, the libraries you may want to use can perhaps demand for a language, like glm is a C++ library. So I suggest to use C++ from this variety.