That's no moon....

No, because absolute size is not what makes a moon a moon. Our Moon is a moon because it directly orbits a planet, not a star. Charon is massive enough relative to Pluto that the former does not directly orbit the latter, but instead they both orbit a common barycenter located between them, making them a binary planetary system.

Does our moon independently orbit the sun?

The barycenter between earth and the moon is within the earths crust. The barycenter between charon and pluto is outside of plutos surface. For those who dont understand, this means that the center of gravity between the earth and moon is INSIDE the earth. So the moon orbits a point within the earth. Not so with the charon pluto system. Both worlds orbit a point in space.

If the moon was to have its own orbital path around the sun, then sure. It would be a planet imo. It's rounded by its own gravity... and it would orbit the sun.

But I guess if we want to the meat of the subject about what defines a planet in the most basic sense, it would be things that make Earth a planet, since we pretty much all agree Earth is a planet. So.. rounded... Orbits the sun... What else would you say describes the Earth?

Gaia and Selene are double planets. Change my mind.