dSsur, dS and dStotal relation By definition, 


  dS=dqrev/T.


Based on my understanding, the subscript rev means that the dq is heat exchanged in a reversible pathway from a state i to another state f, regardless of the actual pathway by which the process proceed. I am always curious why ΔS is defined only for reversible process, it would be good if someone can explain it, nonetheless it is not our main concern here-let's just accept the definition first.

So, similarly,


  dSsur=dqrev,sur/Tsur......(1)


From what I have read, V of surrounding can be regarded as constant, so


  dqrev,sur=dUsur


and since dUsur is a state function,


  dUsur=dqsur


and therefore dqrev,sur=dqsur implying 


  dSsur=dqsur/Tsur......(2)


(a) If V is constant, q=ΔU, so q is a state function at constant V?

(b) Let's stop at equation (1). Suppose we would not like to simplify it further, and since the equation (1) is an intermediate result it must be true, we have 


  dS=dqrev/T and
  
  dSsur=dqrev,sur/Tsur


I suppose dS without any subscript is dS of system. By First Law, 

dqrev= -dqrev,sur

and then if T=Tsur and the chemical reaction has not reached equilibrium (is this scenario possible?),


  dS= -dSsur


By summing up all entropy changes for system and surrounding, we have
ΔStotal=0 for non-equilibrium state? But as I know ΔStotal=0 if and only if surrounding is in equilibrium with system. What's my misconception here?

(c) How can dqsur,irrev=dqsur,rev? Does not make sense for me physically, without looking at the equation with dUsur
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:
$$\Delta S=\Delta S_{sys}-\frac{Q_{irrev}}{T_{surr}}=\left[\int{\frac{dQ_{rev}}{T}}\right]-\frac{Q_{irrev}}{T_{surr}}$$