These are personal computers so it's your choice to have this or other programs arranged to your liking. There is no correct answer to what desktop icons should be there.

As to deleting files from the download folder, IE have a clear download button but without knowing if you are on which version IE I'll have to guess you have the latest. Here's the tutorial on clearing IE download's list.
http://www.sevenforums.com/tutorials/112240-internet-explorer-delete-download-history.html