Is it true that the more desktop icons you have the slower your machine will run ? Also would removing the icons cause a problem with the original file .exe path ?
Ultimately it's probably true, but with today's hardware, the impact should be negligible. Removing a shortcut to an .exe won't affect the program, however the shortcut may pass arguments to the .exe that could make it function differently than clicking the .exe itself (like the path to an ini or log file, or other arguments).
I configure all my programs... all of them to run on start up. I then minimize all of them into the task bar, so when I want to look at something on a particular program. It's already running and doesn't have to load.