Just curious, how are you displaying the featured products on that page i.e. through the admin->configuration->Index Listing, or are you using another method?
If you used that method, another method that might work to get a lot of customization potential would be to edit the includes/languages/english/html_includes/define_main_page.php and add something like
<?php require($template->get_template_dir('tpl_featured_products_default.php',DIR_WS_TEMPLATE, $current_page_base,'templates'). '/tpl_featured_products_default.php'); ?>
As far as the search under the categories is concerned, did you install some custom module? I would look under includes/templates/template_default/sideboxes/tpl_categories.php or includes/templates/CUSTOM/sideboxes/tpl_categories.php and see if you can edit out any search box material in the php code that might be making it display.