Why are there dots on Edges? Why are there dots on some of the edges while other edges do not have dots? 



How do I remove these dots since they affect the subdivide smooth results?:
That means you have zero sized faces over those edges and you mesh topology is messy. Probably the leftover of unfinished operations or the result of some undesired zero sized extrude.

You can try and clean it up by selecting all with A and then with W Menu > Remove Doubles option.