I know that the Foucault pendulum rotation in relation to Earth is a proof that the object is inertial in relation to the distant stars. But what makes them more important than the Earth? Are they an absolute and universal inertial frame? How can we prove that? Please elaborate.
Actually the path of the Foucault Pendulum is not "fixed" (even approximately!) to the "fixed" stars. Unless the pendulum is installed at one of the Earth's poles (as someone has done), then the point of suspension is in constant rotation with the Earth itself. $\therefore$ the pendulum is really not in an intertial frame.
A very good discussion of the forces (real and fictitious) on the pendulum can be found at this UNSW site. The vector that points from the suspension point toward the Earth is in constant acceleration and has a precession period that varies according to latitude.
This animation from the Wikipedia article on the Foucault pendulum may help show how the plane of the pendulum is rotating.
It is not true that the Foucault Pendulum is "inertial in relation to the distant stars". The distant stars are moving in various random directions at various random speeds and are certainly not in the same inertial frame as the pendulum. Our galaxy is rotating, so it can't be used as an inertial frame. The visible universe is expanding and probably accelerating, so it's certainly not an inertial frame.
In Foucault's day the movement of the stars wasn't visible because they are so far away, so it was common to assume that the stars were fixed, and therefore represented a fixed framework that you could use as an absolute frame of reference. Actually that's not a bad approximation, but it is only an approximation.