I read, that the Hull Shader, Domain Shader, Geometry Shader and Pixel Shader can be used optional. So, is the Vertex Shader optional too? If no: What does a basic Vertex Shader look like? Just like a simple pass through? Is the Vertex Shader necessary to tell what kind of datastructure (Van Stripes or Meshes) are used? What can I do, with just the vertex shader? Are the fixed functions working without any help of programming a programmable stage?
See Shader Stages on MSDN:
The vertex shader stage must always be active for the pipeline to execute. If no vertex modification or transformation is required, a pass-through vertex shader must be created and set to the pipeline.
From this you can infer that a vertex shader is absolutely required to write SV_Position but that's all, so the most basic possible vertex shader will just write an arbitrary value to SV_Position; in the pass-through case this just means copying over the input position to output, but (since D3D10+ allows drawing without input buffers) you can set it to any value you like if you wish (this is commonly seen in code for drawing full-screen quads).
If you use compute shaders, or a GPU compute API like CUDA or OpenCL, then you don't need any of the graphics pipeline shaders. Depending on your application, GPU compute may be a better fit for you, e.g. if what you're doing has nothing to do with rasterizing triangles. Compute allows you to simply launch a large block of GPU "threads" running in parallel, and what they do is up to your program.
As for the graphics pipeline, you can't do much with just a vertex shader. You can rasterize triangles to the depth buffer, and that's about it. If you want to do some computation at each pixel of an image, a common approach is to put the computation in a pixel shader, and do a "full-screen pass" by drawing a single triangle that covers the entire image. This requires a very simple vertex shader, but the real work is done by the pixel shader.
Vertex shaders are iirc not optional.
They tend to be used to transform vertices from world to view space and then project them onto a 2d plane (screenspace).
Vertex shaders doesn't know what kind of data structure it is, merely that you're taking in a vertex with some input data.
The vertex shader outputs the projected position and any optional data you want to pass along.