So vertex visibility is totally fixed and working now. The only thing left to do for it, is looking like it's going to be the hardest part so far. That is, checking whether a vertex that is detected inside a face is further away from the camera than the face itself.
That will involve finding the equation for a plane based on 3 coordinates, and finding the intersection between a line and a plane in the 3D system. I found some videos about that last night and it is definitely not trivial looking stuff..