Did you hide your icons via "Right-click desktop > Arrange Icon By > deselect "Show Desktop Icons"?
If so, just reselect the "Show Desktop Icons".
The other easy fix is to install TweakUI from this page (about halfway down the right side of page) http://www.microsoft.com/windowsxp/downloads/powertoys/xppowertoys.mspx
After installing, start TweakUI > select "Desktop" > place a checkmark in the Recycle Bin > click "Apply".
Note - if you made registry changes, or a group policy, then you can change those - but I suspect you would remember those.
Not sure on that one, but if they are folders, you could try right-clicking the folder > Properties > select "Hidden" > click "Apply" > press "F5" to refresh the desktop, which should then hide the folders.
To view the folders later, simply go into "My Computer > Tools menu > Folder Options > View tab > select "Show hidden files and folders" > click "Apply".
Note - I do not use "Avedesk", so I do not know if this will adversely affect the shortcuts you use. It should be harmless to try though, and you can always deselect the "Hidden" attribute on the folder to reset things.