Can the change in the entropy of the surroundings always be obtained by dividing heat transferred by the temperature at which the transfer occurs?
I disagree with the explanation given in the text.  The entropy change of the surroundings is given by the equation presented in your title only for an ideal constant temperature reservoir.  Such an ideal reservoir is characterized by (a) an infinite capacity to absorb heat without its temperature changing and (b) infinite thermal conductivity, so that temperature is uniform throughout the reservoir, including at the interface with the system (where all the heat transfer occurs). So all the heat transfer between the system and the reservoir occurs at the reservoir temperature $T_{Res}$. In this way, all the irreversibility of the process occurs within the system, irrespective of whether the process is reversible, and the entropy change of the system always satisfies $ΔS≥Q/T_{Res}$.  

None of the entropy generation for the process occurs within the ideal reservoir, and the entropy change of the ideal reservoir is thus always $-Q_{sys}/T_{Res}$, irrespective of whether the process is reversible. I should also mention that, although often not stated, the Clausius inequality is supposed to be expressed in terms of the temperature at the interface with the surroundings $T_{Res}$ where all the heat transfer is occurring. 

Typically, all the surroundings reservoirs you will encounter in most thermodynamics texts will be ideal constant temperature reservoirs. 

The description in your post is inappropriate because, if the reservoir has a constant volume (i.e., constant mass), its temperature can change between the beginning and end of the process.  Its entropy change will not be given then by the equation presented.  In addition, the description omits the requirement that the reservoir most have infinite thermal conductivity (so that the temperature in the reservoir is spatially uniform).