Can the search function be turned off from the shopping cart area without taking away the whole left nav panel?

Yes, the search function can be disabled and the left navigational panel can still be shown.

1. Under 'Category' select 'Edit Category'
2. Choose the |Product Search| category
3. In the 'Script for this Category' form field remove the following script: 

 <form action="" method="post"><br /><input name=searchquery type=text size=12><br><input type=submit value="Product Search" style="width: 120px;"></form>