This is a question to those who have AdSense and use the Google search on there site:
why do you have the google search box leave your website when you click submit (aka it goes to http://www.google.com/custom?q=....)?
Google has some APIs which allows you to query there servers and display the results on your website so that you never have to let the user leave your website. That and you can display the search page however you want. Are there some benefits to usign the custom one versus usign there API's (and completly customizing it)? And yes u can restrict the searches to just your website(s) or the entire web, it's really all up to you!