Ok, I better understand what you're after.
First, ANYTHING that you install WILL be installed on your computer, regardless of you installing it on your external hard drive or not. When you installed iTunes on the external drive, it probably still wrote some files to the C: drive, which you don't want to happen.
So, it sounds like what you're really after is a completely standalone application that doesn't require installation OR that won't install anything on the C: drive that you can use to manage your music and your iPod (or at least sync with your iPod). Not every application you run has to be installed first, CPU-Z is one example. Believe it or not, I believe I just found
what you are looking for. I found that on
this site.
I have NOT used any of this software myself so
BE FOREWARNED AND TRY AT YOUR OWN RISK.
Hope that helps.
Peace...