Why does a vertex not become part of the edge it was snapped to? I have tried for tinyCAD, Dynamic Space Bar, Intersect Tool and EdgeTools, Automerge and other stock options, but if I create a new vertex (Shift+D) and constrain it to an axis and I snap it on another edge, it does not merge together and become a lovely piece with that edge. 
Besides, I have a small space to snap the Vertex near to Edge, but not on the edge. And I still couldn't figure out how I to merge the Vertex with the Edge afterwards. 

 

But besides that I really like Blender so far and a lot of things are really well thought out and so I think it is me who is wrong.
To be able to merge this vertex with the edge you have to have another vertex as a part of the edge, to do so, select the edge and by pressing the W key subdivide it once then you can slide the vertex where you want by pressing ShiftV, and now you can merge this vertex with the other vertices.