Help to kick out the anomaly in proving $Q$ = $K_\text{eq}$
In order to compare $K$ and $Q$, you have to define them in terms of the same measures of concentration and the same standard state. The only difference in the definition should be that in order to determine $K$, you are required to use the concentrations or activities of an equilibrium state, while $Q$ may be determined for states that are not necessarily at equilibrium.

please help me to understand why at chemical equilibrium $Q=K_{eq}$ holds

Once you have the definitions match, it becomes fairly trivial. The equilibrium constant is what you observe whenever the reaction has had enough time to go to equilibrium. The individual equilibrium concentrations might be distinct for different starting concentrations, but $K$ comes out the same (if it does not, it means the reaction is not reaching equilibrium). If $Q$ is not equal to $K$, you are not at equilibrium.
I realize this is a bit of a tautology. The explanation why the particular form of the equilibrium expression leads to a value that is a constant requires a knowledge of how chemical potentials (or $\Delta_r G$) vary with concentration. The reaction will proceed in the direction that minimizes $G$, so at equilibrium $\Delta_r G = \frac{dG}{d \xi}$ will be zero.