Siddharth,

The barycentre of the solar system is terribly close to the Sun - sometimes inside its photosphere, sometimes not - so by any reasonable definition of 'orbit around' it's legitimate to say that Jupiter orbits the Sun.

Pedantically, both orbit the solar system's barycentre.

Pretty much non of what you said, is right. Jupiter is not a failed star. Jupiter is approx. 0.1% of the Sun's mass. To start a fusion (of deuterium), i.e. to become a brown dwarf (which is not a star, but more a failed star), it would have to have 13 times of its mass. So it is 1/13 of the mass it would need to be considered anything close to a star.

The statement concerning the rotation of Jupiter is trivial and as made also wrong. No object circles around the Sun's center. All bodies in the solar system move around their common barycenter. As the barycenter depends on the position and mass of all bodies, it is not a constant point, but varies over time. An example for its position can be found on wikipedia (article Barycenter).

Not only Jupiter, none of the free macro bodies in the universe orbits around another macro body. It is physically impossible for a free body to orbit around another moving body, in any type of geometrically closed path. This can be observed by watching a person trying to move around another person running along a defined path. Planets are free macro bodies, central body (sun) is a moving body and circular/elliptical path is a geometrically closed path. Sun and Jupiter are moving bodies. They may orbit about each other not around either of them.

See: http://vixra.org/pdf/1311.0018v1.pdf