So I finnaly got the normal maps to work, but I've some weird problems with alpha channels now. It looks like I'm not able to use normal maps and alpha channels at the same time.
This is a simple cube with a green diffuse map on it and some black spots on the alpha channel. The settings I used for exporting FBX2EDF are 'Normals on recalculate and angle at 90'. Now the normal map works, but I see the other cubes behind the cube, which shouldn't be the case in some spots.

When I use the same cube with the same diffuse, alpha channel and normal map and select 'recalculate' without the angle settings, I get the same results as the picture above.
But when I use the same stuff with Normals settings on 'Flat shading' it looks like it doesn't have any normal map added on the object and the lighting looks weird overall. It looks like the look-through problem doesn't appear though.

When I use the same stuff without any Normals settings it looks exactly like the above one (Normals settings on 'Flat shading'.
Could anyone tell me what's going on here? I've been searching at the GP-bikes and Kart racing pro forums. But I couldn't find the information that I needed. It'd be great if someone from GP-bikes or Kart racing pro could help me out, since I don't think Piboso has a hell lot of time to explain everything I do wrong.