Sun and planets orbit each other Do not the planets and the Sun revolve in orbits around each other and the shape of the orbit depends on where the center of gravity of the system is? The greater the mass of the Sun, the closer the orbit approximates a perfect circle.
 A: No. The shape of the orbit, i.e. how elliptical it is, does not depend on the relative masses of the two bodies.
All objects in the solar system orbit around the centre of mass of the solar system. For obvious reasons, namely that the Sun contain far and away most of the mass of the solar system, the centre of mass of the solar system is quite close to the centre of the Sun.
The shape of those orbits is determined by the total energy of a body (i.e. the sum of the potential and kinetic energies) and the way that the energy is shared between kinetic and potential energy. The potential energy is always negative by definition. Bound, elliptical orbits have a negative total energy. Circular orbits are a special case of elliptical orbits and have a potential energy that is exactly twice the total energy. i.e. if the total energy is $-E$, the potential energy is $-2E$ and the kinetic energy is $+E$. For stable elliptical orbits this is true on average, but at any instant in time, the split between potential and kinetic energy can be different and changes as the body moves around in its orbit.
A: To say that the orbit becomes more circular the greater the Sun's mass is not true. Instead, the eccentricity (i.e. how much the shape of an orbit varies from being circular) is governed by a couple of factors.
If you have a planet orbiting about the Sun with a mass much less than that of the Sun, and you know the following for an instantaneous point in the orbit:
orbital radius, $r$
radial velocity, $v_r$
tangential velocity, $v_t$    
then, the eccentricity is given as follows:
$$e=\frac{r}{GM}\sqrt{\left(v_t^2 - \frac{GM}{r}\right)^2 + \left(v_r v_t\right)^2}$$
Therefore, in order to get a circular orbit, the planet needs to follow two conditions.
$v_t^2 = \frac{GM}{r}$ which is equivalent to $\frac{mv_t^2}{r} = \frac{GMm}{r^2}$
$v_r = 0$
A: Yes, objects with mass all attract to each other and move each other of course, except that the star doesnt change it's theoretical orbital shape depending on it's mass, it probably just experiences small tidal forces that aren't much bigger than it's own centrifugal forces. The oscillation of the star position is a complex dynamic based on it's surrounding stars, activity, rotation, and fluidity that all work together. if the sun was twice it's current mass, the oscilaltions of its theoretical centre probably wouldnt be more curcular mathematically.
A: When two object orbit each other, the shape of their orbit is independent of their relative mass. In fact they will each have the same shape of orbit, but scaled by the mass of the other. So if you have two objects of mass $m$ and $2m$ respectively, then the former will have an orbit (circle or ellipse) that is exactly twice as big as that of the other. 
When the orbit is an ellipse, the barycenter will be the focus of the ellipse; when the orbit is a circle, the barycenter is the center.
