Does a natural affect the value of the stone?

Yes! Consider this example: You are shown two diamonds that have been certified as identical by a major gem lab (i.e. weight, color, clarity, proportions, etc.). One diamond has two naturals on either side of the girdle that are easily visible under a microscope, the other diamond is completely polished and without any naturals. Which diamond would you choose assuming the price was the same?

Another way to detect naturals, is to purchase a "certified diamond". Every certified diamond comes with a drawing which depicts the location of all internal inclusions marked in red. Nicks and naturals that break the surface of the diamond are identified with green marks. Below is a sample plot of a diamond with naturals. The red marks show the location of all inclusions visible from the top and bottom views. The two green lines that extend from one end of the girdle to the other are naturals. Notice that they are drawn from the bottom plot view because these naturals are not visible from the top.

Bellman's CoolSchool
Extra Facets
An extra facet is one that is not part of the traditional 58 facets that make up a Round Brilliant Cut diamond. Like naturals, they are usually located below the girdle in the pavilion. Quite frequently, an extra facet is created by polishing out a natural in the stone (normally having no or very little effect on the grade of the diamond). However, some extra facets are very large and flatten a significant area of the pavilion reducing the stone's brilliance.

Note the extra facets in the diagram are marked in black
vs. green for naturals.