dSsur, dS and dStotal relation
In thermodynamics, the surroundings are typically treated as an ideal constant temperature reservoir for which $\Delta S$ is always equal to $Q_{surr}/T_{surr}$, irrespective of whether the process that the combined system and surroundings experiences is reversible.  As such, the surroundings is assumed to have an infinite heat capacity and thermal conductivity, and to not change significantly in temperature (its temperature is always $T_{surr}$), no matter how much heat it receives or rejects.  So it always presents the temperature $T_{surr}$ to the system at its interface with the system.  And, all the entropy generated in any process is confined to the system. 

In evaluating the entropy change for system and surroundings, the system and the surroundings must first be separated from one another, and then each must be separately subjected to a reversible path.  These separate reversible paths do not have to match up in any way whatsoever.  For the surroundings, the final state is that which was obtained with the irreversible path when $-Q_{irrev}$ was transferred from the system. So the change in entropy for the surroundings is $-Q_{irrev}/T_{surr}$.  And the change in entropy for the combination of system and surroundings is then given by:
$$\Delta S=\Delta S_{sys}-\frac{Q_{irrev}}{T_{surr}}=\left[\int{\frac{dQ_{rev}}{T}}\right]-\frac{Q_{irrev}}{T_{surr}}$$