Here's how I do a "clean" profile:
Start Firefox. (First off, rename your original profile back)
Type "about:profiles" in the address bar
Click "Create a New Profile"
Give it a name; e.g., "Clean Profile" and let the browser build it
Under the newly created profile, click "Set as Default Profile" (do not click "Launch Profile in New Browser;" it won't be entirely "clean")
At the top, click "Restart Normally"
Do your testing
To switch back, do steps 2, and 5 and 6 again with your usual default profile
You can keep the "Clean Profile" for later tests of this sort.
If it works with a clean profile, you can create yet another profile, then start adding back your add-ons, etc. until you find out what's causing the problem with Skype.com.