To make a new face from existing vertices, select them and use Vertex->Create_ordered surface. (To have it working like you said would not be possible - because the vertices may be in different objects, and surfaces can only use vertices from one object)
To make a vertex in the middle of a surface, use Surface->divide.
There are some new edge commands in 3.7 but deleting an edge is not in there yet. It's high priority though.
>a button I can press to make all of my models look good.
Now this would be a great feature! I'm not sure how it would be implemented though
Andy