Why are planetary orbits elliptical and why do bodies in a solar system orbit the center of mass and nit the star itself?
Planets orbits are defined by conservation laws.
Johannes Kepler discovered by observation that planets follow elliptical orbits. A few decades later Isaac Newton proved that by applying the law of conservation of energy that a planet's orbit is an ellipse.
When two bodies orbit around each other, they both always orbit about the centre of mass. This centre of mass is called the barycentre. The Moon doesn't orbit around the Earth. In fact both the Earth and Moon orbit around the Earth-Moon Barycentre (EMB).
When it comes to something more complex like the solar system a similar principle applies. None of the planets etc actually orbit around the Sun. In fact the Sun, planets, asteroids, comets and other bodies all orbit around the centre of mass of the solar system which is called the Solar System Barycentre (SSB).
The SSB is in constant motion and can be anywhere from near the centre of the Sun to over a Su radius outside of the Sun. So, everything in the solar system is orbiting around a point which is in constant motion.
The diagram shows the path of the SSB over several decades. The points where the SSB is furthest from the Sun occur when the planets are aligned.