Skype stays in my taskbar even when I sign out, even when I right-click on the item and select close window. How do I exit skype altogether, without having to restart my computer or kill the application in task manager?
If you don't want that Skype will run in the background and completely sign out, then from the main menu (located in the top of your Skype window) choose Skype --> Sign Out
This will sign you out from Skype.
Please visit the link below to see the available Quit option in the recent Skype versions.
In Skype, Tools >> Options >> Advanced >> Advanced Settings >> Connection >> uncheck the enable uPnP and click Save.
Restart your computer and try again.