There are these strange black lines which are covering the model and I do not have a clue, what I did wrong. This hasnt happened to the other models I made and it carries through Unity3D as well. If anyone could help me with this issue, I would really appreciate it.
You need to space your UV islands with room for bleed. Bleed is overpainting (just like in printing). In the case of 3D texturing, the anti-aliasing of the textures will pull in the color of the surrounding texture a small amount. The trick to fix that is to overpaint. In your case the background of your texturemap is black, so the lines which show around all your UV islands are black too.