Right click on the bar at the top of the search pane and click remove search or close search from the drop down menu. Also you can click the red X on the bar above the search. but this does not always work for me. You can have a number of searches running side by side. The default is 5. If you open a further one, one of the others will dissapear/be closed.
BTW the star quality is to indicate the quality of the download rather than the quality of the file. Personally I take no notice of the number of stars, but you cannot close the column because of the other things it shows.
In a session Limewire keeps a history of your searches, and if you have autocomplete enabled this will autocomplete search terms you re-enter. You will also get the option to cancel the search history.
Restarting Limewire will alo clear the history, and afaik that is all the history is
The list I explained how to remove, is the Search results. From this list you chose what to download and it appears in the download pane which appears under the search pane, and you can watch the download progress.
I hope I understood your question correctly and that this helps
----------------------EDIT
JEEZ have I just answered a multiple post yet again? I see you posted this question several times <SIGH> I'm done for today.