ArcGIS is particularly poor at accessing data across the network. It can be dealing with quite a large number of smallish files or some very large files, depending on what it's doing. If you have large datasets then, in order to improve performance and not overload the network, copy, using arcCatalog, all the data layers that you need to do your processing onto the local drive of your machine( in C:\Workspace) into a clean folder. Do your processing there and then copy back the results that you wishes to keep, again using ArcCatalog and NOT windows explorer, back onto your M: drive. You will find that everything goes much faster, even if there is an initial overhead in doing the copying.
If using data on the network drive, do not use a mixed case path, it will give unpredictable results or just crash the software on occasion.
eg M:\Dissertation\My Project\Raster1 - should be M:\dissertation\my_project\raster1
To access the Zonal Statistics and Raster Calculator tools: start ArcMap [on the machine in the Drummond Street Labs( basement( room 1.26 or 1.23))]. Go to the Tools/Extensions and switch on the Spatial Analyst extension. This setting should stick for future sessions on different machines as it is held in the ESRI configuration in your profile.