Does the Earth revolve around the Sun? I am aware of this Phys.SE question: Why do we say that the earth moves around the sun? but I don't think this is a duplicate.
In a binary star system, where the masses of the 2 stars are not so different from each other, can we say that each star revolves around the other?
If yes, then couldn't the Sun-Earth system be an extreme case of such a system?
Therefore, strictly speaking, can we argue that the Sun revolves a tiny bit around the Earth as well?
 A: In binary systems, each object is so affected by the others gravity that they have significant orbit. The sun has so much inertia that the earth's pull barely moves it, but the earth certainly revolves around the sun. In the reference frame of the Earth however, the Sun does revolve around the Earth.
A: Both the the Earth and the Sun orbit around the solar system barycentre. This is defined as the centre of mass of all the bodies in the solar system. Because the Sun contains the vast majority of the mass of the solar system then the barycentre is very close to the Sun. The picture below, from the wikipedia entry on the solar system barycentre, has the barycentre stationary in the middle of the picture, and illustrates schematically the situation when one body is a lot more massive than the other (though is not to scale for the Earth-Sun system!).
The Sun executes a complicated orbit around this point (also illustrated here), pulled by the motions of, primarily Jupiter, but all the other planets also make a smaller contribution.
It is this "reflex motion" of a star, caused by planets in its solar system, that allows the detection of exoplanets by the doppler method. The orbit of the Sun around the barycentre would cause it to appear to a distant observer to be periodically redshifted and blue shifted with an amplitude of about 13 m/s, with a period of around 12 years (the orbital period of Jupiter). In isolation, the Earth would only cause the Sun to orbit the barycentre of the Earth-Sun system with a speed of 7 cm/s (which is one reason that finding Earth-like exoplanets is very difficult).

A: There are at least 8 more planets in the solar system, besides the Earth (and some more were discovered). When Copernicus decided to place the Sun in the center of the solar system, instead of the Earth, that was mainly because this arrangement simplified drastically the form of the orbits of the other planets. With the Geocentric model of the solar system those orbits appeared as very complicated.
Later on, the Kant-Laplace theory for the solar system, strengthened this Heliocentric configuration.
