Can a 10 year old gymnast that uses an instrumental version of a commonly known song as her floor music be in violation of copyright violation?

It is impossible to give categorical answers to "fair use" questions. The determination is always extremely fact-instensive. The Copyright Act gives the copyright owner (i.e., the owner of the composition copyright) the exclusive right to perform the work publicly. While the limited and non-profit nature of the use made by the girl are factors that tend to favor her, remember that "fair use" looks primarily at the economic injury, if any, to the copyright owner. In this case, for example, we do not know whether the copyright owner regularly licenses the work for public performances such as these and, accordingly, whether those royalties are being avoided through this use. When in doubt, ask for permission. Such permissions are often granted on surprisingly reasonable and favorable terms.