No, the Earth does not rotate around the Moon, and our orbit around the Sun is not significantly affected by the Moon's presence. The Earth-Moon system is not considered a binary planet system.
The Moon is Earth's natural satellite, meaning it orbits around the Earth. The Earth-Moon system can be thought of as a two-body system, where both bodies (Earth and Moon) orbit around their common center of mass. However, the center of mass of the Earth-Moon system, known as the barycenter, is located within the Earth but not at its exact center. This is because the Moon is much less massive than Earth.
As a result, the Earth does experience a slight wobble or "lunar libration" due to the Moon's gravitational influence. This effect causes small variations in Earth's rotation and the position of the Earth's axis over long periods. However, these variations have a negligible impact on Earth's orbit around the Sun. The orbit of the Earth around the Sun remains largely stable and is not significantly affected by the Moon's presence.
It's worth noting that Pluto's status as a planet has been the subject of debate among astronomers. In 2006, the International Astronomical Union (IAU) redefined the definition of a planet, and Pluto was reclassified as a "dwarf planet." Size alone is not the determining factor for classifying an object as a planet, as other factors, such as the object's orbital characteristics and its ability to clear its orbit of other debris, are also considered.