RadSearchBox does not support the functionality to click on it and see all the items because it actually does not have items collection. If you are looking for such functionality you will have to use RadComboBox as in this on-line demo for example.
As for your second question- yes of course you can use it as for some other custom search by using onButtonCommand event.
Hope this information will be helpful.
If you want to get updates on new releases, tips and tricks and sneak peeks at our product labs directly from the developers working on the RadControls for ASP.NET AJAX, subscribe to the blog feed