You can completely customize your Customer Portal Home screen by adding Widgets and Sections based on your requirements. You can also choose to hide or show the default widgets and sections in the portal. To enable these customizations, Goto Setup > ...
Yes, you can customize your search. Goto Setup >Tab Settings >Search Layout >select the module/tab >edit and select the fields that you would like to search. Please Note: You cannot select more than 6 fields.
Yes, you can re-name Tab as per your organization requirement. Goto Setup >Tab Settings > Rename Tabs. Now, edit a particular module to change the display name of a tab. For more refer Tab Settings > Rename Tabs.