Why is ΔS the same regardless of whether a process is irreversible or reversible?
In an irreversible process, we can't use the integral of dQ/T to calculate the change in entropy between the initial and final states of the system, because there is entropy being generated within the system (as a result of mechanical energy dissipation to internal energy) which the integral does not account for.  However, since entropy is a function of state, we can focus on the exact same initial and final states of the system, and devise a path between these two states that does not involve dissipation of mechanical energy, and calculate the integral of dQ/T for this alternate path.  Such a path is a reversible path, and the integral along this (or any other) reversible path gives us the change in entropy.