How do you proof that $F = - kx $? And why is there (-) on the formula(?)
No. There are two points of view on this. First, it's a definition of what a simple harmonic oscillator/ideal spring is. Second, it's an empirical formula that is approximately correct in many circumstances. It is literally the same thing as searching for the minimum of a function by setting the derivative of the function equal to zero, and then approximating the function there with a parabola.
The minus sign is there so that the parabola (in potential energy) is pointed upward instead of downward. In other words, "uphill" is any point away from $x=0$, so the force pushes the object back toward there.
$F = -kx$ is the definition of a harmonic oscillator. It's not something that's proven. Most real systems can be reduced to a harmonic oscillator for sufficiently small displacements from the equilibrium; the proof of that involves Taylor expansion.
There's a $-$ in the formula because the force is restoring - it always attempts to shift the object back to the equilibrium point.