Those actions are available in Solidworks Explorer, located by default in the Solidworks 201x folder within Start Menu -> All Programs. That is what is represented with these icons in right click menu. It will also require definitions (setup) of which locations contain Solidworks files, or else it won't properly Rename a file including all references to it everywhere that it is used. Those locations are defined in Solidworks Explorer (Options -> References/Where Used), but I think it also carries a location defined from within Solidworks.
Furthermore, it appears that there is something wrong (or at least not standard) with your file extension of .SLDPRT files. Its file icon is not the Part icon, and its extension description appears as, "SLDPRT-bestand". It appears that someone has redefined file extensions on your computer since defaults were created with the installation of Solidworks. There must be a reason for this: if someone else supports this PC for you, then ask your administrator why this was set and if it can be returned to defaults. A simpler fix, if there is no apparent reason for messing with file extensions, is to "Change -> Repair the individual installation (on this computer)." from within Control Panel -> All control Panel Items -> Programs and Features. If that doesn't work, un-installation and re-installation should definitely recreate file extensions on that computer along with all the rest of Solidworks.
Once your file extensions are back to defaults, the Windows Explorer right click menu should work for parts like the other entities. I strongly recommend setting References/Where Used locations so that a renamed Part does not have its references broken in assemblies or drawings where it is used.
Thanks for the information. Now the problem has been solved.