Will the meaning change if I say "skunk's butt rug"?
Yes, the meaning will change if you say "skunk's butt rug", because this places the subject of the skunk in the possessive form. So this could mean that the butt rug belongs to the skunk. However, "skunk-butt rug" is hyphenated, which joins the words "skunk" and "butt" together, and the combination of them to forms one word. The term "skunk-butt" is used here to describe the rug in the sentence, showing that the rug is made from a skunk butt, but the rug in itself does not belong to the skunk in this case.