Finally! I have found the problem :-)
GDB immediately told me that it could not load some DLLs, and segfaulted.
The non-existing DLLs appears to be leftovers, from an old version of the surveillance application called "Digital Guardian".
It is some crap-ware that injects itself deep into the OS, and monitors all sort of stuff, like file transference.
It is very likely that a white-list exists, and this could be why all our "normal" programs was working perfectly.
I have yet to find a way to fix it, but now I can at least blame a borked installation of "Digital Guardian".
In any case, it is reason enough to justify giving any user with the problem a new PC. Which is likely to be the solution.
Thanks to everybody for the input, and especially to you Martin, for pointing me in the right direction. Much appreciated.