I have a question. Is it possible to create points which is automatically created on existing polygon? Just like in the following picture
No I do not believe you can automatically create a single vertices in the midpoint of a side of a polygon.
However, it can be done in a more complicated way. convert you polygons to polylines.
convert the polylines to points and then again generating hte midpoint.
then create polylines from the points and the polygons
Are those points at the starting vertex of the polygon? You could use a group feature template to automatically create a point at the starting vertex of a polygon. After you set it up and draw a polygon it automatically puts a point from a different feature class at the very first vertex. In the picture below, I have two feature templates for my Field Notes layer, one to draw a regular polygon and one to draw a polygon, but also automatically put a blue meter (M) at the starting vertex. Follow the link above for instructions to create it.
Technically, this is only for creating new features, not existing ones like you asked. That would require a GP tool to find the correct location on the perimeter of the polygon, then draw the point. Can the point be in the center of the polygon? There is a tool called Feature to Point that will create points in the geometric center of all of your polygons and the new points will have the same attributes of the polygons.
Regarding the group feature templates above: you can actually do a mixture of any geometry. Originally, the data in the picture is meant to draw a water meter (point) at the end of every water lateral I draw (line). Otherwise, that would be two separate operations. First draw the water lateral, then draw the meter. The group template creates the meter as soon as I finish drawing my lateral.
Retrieving data ...