Does the Earth actually rotate around the sun? Why does one say that the Earth rotates around the sun? There should be no preferred reference frame and from an observer on the sun, the sun will rotate around the Earth. 
 A: People say the Earth rotates around the sun and not vice versa because a reference frame attached to the center of the Sun more closely approximates an inertial reference frame than a reference frame attached to the center of the Earth.  
Yes, there is the issue of the path of the other planets, but even if the Earth were the only planet, we would still say it rotates around the sun, and not vice versa.
There's nothing wrong with, or arbitrary about, preferring to describe a system from the point of view of an inertial frame rather than a non-inertial frame; they are qualitatively and experimentally distinguishable.
A: 
There should be no preferred reference frame and from an observer on the sun, the sun will rotate around the Earth.

Aside: You have the sense backwards. From the perspective of an observer on the Earth, it's the Sun that appears to rotate around the Earth, and at three frequencies, once a day, once a year, and once every 26,000 years. An observer on the Sun would see the planets rotating about the Sun rather than the Sun rotating about the Earth. Moreover, this heliocentric observer would see a much simpler set of frequencies than the complex motions needed to describe motion in the solar system from a geocentric perspective.
From an inertial perspective, those three rates are due to the Earth's rotation, the Earth's orbit about the Sun, and the precession of the equinoxes. So why didn't the ancients see things that way? After all, Aristotle, Hipparchus, Ptolemy, and others were extremely brilliant scientists, on a par with the best of today's scientists.  Stepping back from this geocentric perspective took a great amount of thought and observations. As far as those ancient astronomers were concerned, the Sun, the planets, and the stars were just little lights. How could a tiny little light have any effect on motion?

Why does one say that the Earth rotates around the sun?

This is a good question, particularly in light of relativity theory. After all, all reference frames are equally valid.
That doesn't mean there aren't "preferred" frames of reference. One definition of a preferred frame of reference is whatever frame makes the equations of motion the simplest, makes those equations of motion easiest to solve, and yields the most accurate solutions. This means that the preferred frame of reference is context-depedendent. Here's a challenge: Use a non-rotating, heliocentric frame to yield an accurate weather forecast of the weather in London. Good luck! The preferred frame for weather forecasting is exactly the same frame those ancient scientists used. That means fictitious forces such as the centrifugal force, the Coriolis force, and tidal forces. So be it. All frames of reference are equally valid, after all. Some are just more accurate.
What about describing the motion of a satellite in orbit about the Earth? Once again, the preferred frame (the most accurate) is not a heliocentric-based frame. It's an Earth-centered frame in which the Earth rotates once per sidereal day. The gravitational acceleration of the Earth toward the Moon, the Sun, and the planets is better treated as a fictitious force that aerospace engineers and solar system astronomers typically label as "third body forces".
The question at hand is not about the weather or artificial satellites. It's about the behavior of the Earth and Sun, and by extension, the other planets, the dwarf planets, the asteroids, and the comets that orbit about the solar system. In this context, choosing an Earth-centered frame doesn't make a bit of sense. A heliocentric (Sun-centered frame) makes much more sense, and a barycentric frame (solar system center of mass) makes even more sense. This is by far the preferred frame for explaining the behaviors of the objects that comprise the solar system.
A: I prefer to think of it that the Earth and Sun actually orbit around their combined center of mass, which just so happens to be very deep inside the sun.
The same can be said for the Earth-Moon system. 
