What's the actual path of the planets?

This is an interesting question, since it raises the problem of the reference frame where Kepler's laws are true, which is often neglected.

As a consequence of Newton's laws, in the inertial reference frames where the center of mass (c.m.) is fixed (there is a triple infinity of them, differing only with respect to the position of the c.m.) both planet and Sun describe an elliptic motion having the center of mass as one focus of the ellipse. The two ellipses are similar, with a rescaling factor equal to the planet/Sun mass ratio.

In every other inertial frame, the elliptic motion is combined with a uniform translation, therefore, in such systems, no closed orbit exist anymore.

There are two additional reference frames where the orbit is an ellipse. Both are non-inertial. One is the non-inertial reference frame where the Sun is fixed. You correctly noticed that the Sun is non-stationary. But this is true in any inertial frame. If one picks precisely the non-rotating, non-inertial system where the Sun is fixed, it stays forever at the position of one focus of the elliptic orbit of the planet. Similarly, one could sit on the planet without rotations, and in that system the orbit of the Sun would be again an ellipse like the one of the planet, with the planet at one focus position.

In conclusion, there is not the actual path. Shapes and properties of the orbits are not invariant with respect to changes of reference.

Take a basket ball, how would you map it mathematically? With an equations describing a sphere, where the center of the sphere is the center of mass of the ball, no?

Throw it to the basket, would you still call it a sphere?

The difference with the elliptical trajectory of a planet around the sun is that it is not solid. Still it is a mathematical mapping of the trajectory where the sun is in one of the focuses. The mathematics does not change if the observational reference system changes. The whole ellipse will be describing an additional motion, but the description of sun-planet will be always an ellipse with a sun as the focus. The trajectory of the planet itself will be different for different reference frames but the ellipse mapping will always be there

with the sun as one of its foci

That's probably the source of your confusion. The focus (one of them) is the place of the center of mass of the system (also called barycenter). The elliptical orbit is around that CM of the entire system.

But because usually the star takes up the large part of the mass (well into the high-90%s for single-star systems such as the Sun), then the center of mass ends up very close (often inside) the star, but not at the center of the star. From afar (enough that a pointlike star is a good enough approximation), it resembles that the star is on one of the foci of the ellipse.