Yes, you can re-order Tabs as per your organization requirement. Goto Setup >Tab Settings > Organize Tabs. Now, edit a particular module to change the display name of a tab. For more refer Tab Settings
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.
To hide few tabs/modules for your organization, Goto Setup >Tab Settings >Organize Tabs >remove the unwanted tabs from the display and save the settings.