The forward reaction is endothermic (ΔH = +57 kJ mol⁻¹). Heating shifts the position of equilibrium right (browner) and raises Kc. Cooling shifts it left (paler) and lowers Kc. Temperature is the only change that alters Kc.
There are 2 moles of gas on the right, 1 on the left. Compressing deepens the colour at once (higher concentration), then the position shifts left toward fewer moles. Kc is unchanged — only Q is disturbed, then returns to Kc.
Inject NO₂ and the system opposes the increase, shifting left to remove it. Inject N₂O₄ and it shifts right. Either way the system moves until Q returns to the same Kc for that temperature.
The expression [NO₂]²/[N₂O₄] is the reaction quotient Q at any instant. After a disturbance Q ≠ Kc, so the reaction shifts. When Q = Kc again, the system is at dynamic equilibrium and the colour stops changing.