Precession of the Equinox
The sun does not orbit another star. A star, even a very small brown dwarf, would have been seen, if not in visible light then in the infra-red by the WISE survey.

However there is no mystery about precession. The Earth is non-spherical, it is wider around the equator, and because the Earth is tilted with respect to the ecliptic, there is an asymmetry in the force of the sun, and the moon, on the Earth. This force pulls the axis of the Earth's rotation in a circle. The basic maths of this is not so hard, and is in the Wikipedia page on Axial precession. This explanation of axial precession was known to Newton. There is nothing controversial about it. 

The sun does orbit the galaxy it takes about 230000 years to complete one orbit. This has nothing to do with axial precession.

The page you link to is worthless. It offers an "alternate explanation" of a phenomenon that is very well understood, and pretends that there is some mystery, where there is none. It the uses this to justify a lot of astrological nonsense. 

To repeat. 


There are not "many theories" explaining Axial precession. 
Axial precession is a well understood phenomenon, due to the Earth not being a sphere and the axis of the Earth not being perpendicular to its orbit 
The sun does not orbit another star. We know this because we could have seen it.
No astrology based on the presence of another star is valid.