Definitely follow caperjack's advice concerning the the Spyware/hijackware removal utilities he mentioned.
In terms of restoring the IE icon on your desktop: open Window's Explorer (not Internet Explorer) and navigate to your Program Files->Internet Explorer folder. Right-click on the IEXPLORE.EXE file, go to "send to", and choose the "Desktop (as shortcut)" option. That will put a new IE icon on your desktop; click it and see if IE fires up from there.
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Wombat At Large
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since Dec 2003