Can I draw dog fur with pencil and then go over it with colored pencils?
You can certainly add color into a monotone image (like you would have with a graphite pencil drawing). You are correct that there is the risk making the drawing messy or muddy though, so use (workable) spray fix to protect the base drawing before adding light shading with the colored pencils (or water color, etc.).
However using color when drawing in the style of the video, as well as in any added shading, will give you cleaner colors overall. So if you want the fur itself to have color it would be best to work with colored pencils for the line work as well as any tone work you might add after the base drawing. Try using a few different colors when drawing the fur to add more realism; check out some close up photos of fur to get the idea.
In general when doing works in color black becomes a very strong visual element compositionally. So if you do a black and white drawing first any colors you add later will be secondary to that base image. If you do a drawing (or painting) in color any black you use will really stand out. Finding the right balance between the two is the key.