How are you making the Desktop links?
I have just tried creating a link by the simplest possible method: Rt click on the desktop, select Createnew->File->Link to application, select 'application' type in name of executable (I just used 'kcontrol' as a test) & close. Icon is still the default gear but click it & up pops control center.
For mc you have to go to advanced options & select 'run in terminal'.
Mozilla worked also.
All of the links were owned by user knoppix.