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In reverse order ....
Tools > Internet Options > Privacy and uncheck 'Turn on popup blocker'. However you can leave this checked and hold down the Ctrl + Alt keys when clicking again on the link.
For more information IE > Help - type in pop
in the Index keyword box.
When I double click on the Internet Explorer Icon it doesn't open
Which one - Desktop or Quicklaunch?
I can open it by right clicking the icon and clicking "Run As"
That menu is only available Quicklaunch so I guess this is the icon you are referring to and that requires only a single click.
Right click > delete removes that icon from Quicklaunch - create a new one by left click, hold and dragging the IE icon from the Desktop to the Quicklaunch bar.
See if that works.
Does double clicking always work on other desktop icons?